E∞-Comodules and Topological Manifolds

A Dissertation presented

by

Anibal Medina

to

The Graduate School

in Partial Fulfillment of the

Requirements

for the Degree of

Doctor of Philosophy

in

Mathematics

Stony Brook University

August 2015 ii

Stony Brook University

The Graduate School

Anibal Medina

We, the dissertation committee for the above candidate for the

Doctor of Philosophy degree, hereby recommend

acceptance of this dissertation

Dennis Sullivan - Dissertation Advisor Mathematics

Alexander Kirillov - Chairperson of Defense Mathematics

John Morgan Mathematics and SCGP

Gregory Brumfiel Mathematics, Stanford University

This dissertation is accepted by the Graduate School

Charles Taber Dean of the Graduate School iii

Abstract of the Dissertation

E∞-Comodules and Topological Manifolds

by

Anibal Medina

Doctor of Philosophy

in

Mathematics

Stony Brook University

2015

The first story begins with a question of Steenrod. He asked if the product in the cohomology of a triangulated space, which is associative and graded commutative, can be induced from a cochain level product satisfying the same two properties. He answered it in the negative after identifying homological obstructions among a collection of chain maps he constructed. Using later language, his construction could be said to endow the simplicial chains with an E∞-coalgebra structure. The second story also begins with a question: when is a space homotopy equivalent to a topological manifold? For dimensions greater than 4, an answer was provided by the work of Browder, Novikov, Sullivan and Wall in surgery theory, which in a later development was algebraically expressed by Ranicki as a single chain level invariant: the total surgery obstruction. After presenting the necessary parts of these stories, the goal of this work will be to express the total surgery obstruction associated to a triangulated space in terms of comodules over the E∞-coalgebra structure build by Steen- rod on its chains. a los ´arboles en que estas ideas se escribieron Contents

Introduction 1

1 Simplicial sets and S-coalgebras 3 1.1 Operads, coalgebras and comodules ...... 3 1.2 The operad S ...... 7 1.3 Simplicial sets and S-coalgebras ...... 16

2 Abelian sheaves and S-comodules 27 2.1 Sheaf theory of posets ...... 27 2.2 Simplicial complexes and Ranicki duality ...... 39 2.3 Topological manifolds and S-comodules ...... 48

A Categorical Background 57

References 65

v Introduction

The goal of this dissertation is to relate the theory of algebraic surgery de- veloped predominantly by Andrew Ranicki, with that of E∞-structures on chain complexes. Steenrod’s construction of higher chain approximations to the diagonal inclusion has been encoded, by several authors, as a functor from simplicial sets to their normalized chains enriched with the structure of a coalgebra over an E∞-operad. The first section of Chapter 1 presents the definition of an algebraic operad as well as the less common notions of coalgebra over an operad and comodule over one such coalgebra. The second section presents the specific E∞-operad S related to Steenrod’s construction following the work of McClure-Smith [25], Berger-Fresse [4] and others. In the last section of Chapter 1, the first of the two main technical results of this dissertation is presented as Theorem 1.3.5. It has as a corollary that the category of based ordered simplicial complexes embeds as a full subcategory into the category of S-coalgebras. Similar results have been obtained at the level of the homotopy category by Mandell [20], Smirnov [38], Smith [39] and others. The first section of Chapter 2 revisits the theory of sheaves and cosheaves over posets, see [8], [37] or [14] for other sources. It uses the connection be- tween posets and Alexandrov topological spaces, extended in Lemma 2.1.5 to a duality preserving equivalence, to emphasize the symmetry between sheaves and cosheaves over posets. This section closes with some homological algebra of such sheaves and cosheaves with values in an abelian category. In the sec- ond section of Chapter 2, the sheaf theory developed in the previous section is specialized to posets associated to ordered simplicial complexes. The notion of tensor product of functor is used to define the Ranicki duality functors of complexes of sheaves and cosheaves, whose geometry is made apparent by the pair subdivision sheaf and cosheaf. The pair subdivision sheaf is also used to define the visible symmetric complex of a regular pseudomanifold,

1 see Construction 2.2.18, which plays a central role in the application of the theory to manifold existence and uniqueness problems. The third section of Chapter 2 contains, as Theorem 2.3.4, the second main technical result of this work. It states that the category of complexes of sheaves over an ordered simplicial complex X with values in Ab embeds, as a full differential graded subcategory, into the category of comodules over the S-coalgebra C•(X). This theorem is used to relate the algebraic surgery theory of Ranicki with comodules over E∞-coalgebras. In particular, Theorem 2.3.13 and Theorem 2.3.15 provide existence and uniqueness statements for ANR homology man- ifold structures and topological manifold structures on the homotopy type of a Poincar´eduality regular pseudomanifold, in terms of comodules on its S-coalgebra of chains.

2 Chapter 1

Simplicial sets and S-coalgebras

Convention. The term chain complex will be reserved for a homologically graded differential graded abelian group. The category of chain complexes, 0 denoted by Ab•, is enriched over itself, i.e. HomAb• (C,C ) ∈ Ab• for every 0 pair C,C ∈ Ab•. In terms of this enrichment, chain maps correspond to 0-degree cycles, while chain homotopy equivalent morphisms correspond to homologous chains.

1.1 Operads, coalgebras and comodules

In this section, the definition of an algebraic operad is presented as well as the less common notions of coalgebra over an operad and comodule over one such coalgebra.

Definition 1.1.1. (Operad [22]) An (algebraic) operad consists of a collec- tion of chain complexes O(n), n ≥ 0, a collection of chain maps

γ : O(k) ⊗ O(j1) ⊗ · · · ⊗ O(jk) → O(j1 + ··· + jk), a chain map η : R → O(1) and an action of the symmetric group Σk on O(k) satisfying the following conditions. Pk O1: (Associativity) The following diagram commutes, where s=1 js = j, Pj r=1 ir = i, gs = j1 + ··· + js and hs = igs−1+1 + .. + igs for 1 ≤ s ≤ k:

3 k j j O
O
γ⊗id O
O(k) ⊗ O(js) ⊗ O(ir) / O(j) ⊗ O(ir) s=1 r=1 r=1 γ

 shuffle O(i) O γ  k js k  O O
 O
O(k) ⊗ O(js) ⊗ O(igs−1+q) / O(k) ⊗ O(hs) . id ⊗(⊗sγ) s=1 q=1 s=1

O2: (Unit) The following diagrams commute:

=∼ =∼ O(k) ⊗ Rk / O(k) R ⊗ O(j) / O(j) 8 8 k η⊗id id ⊗ η γ γ   O(k) ⊗ O(1)k, O(1) ⊗ O(j).

O3: (Equivariance) The following diagrams commute, where σ ∈ Σk, τs ∈

Σjs , the permutation σ(j1, . . . , jk) ∈ Σj permutes k blocks of letters as σ permutes k letters, and τ1 ⊕ · · · ⊕ τk ∈ Σj is the block sum:

σ⊗σ−1 O(k) ⊗ O(j1) ⊗ · · · ⊗ O(jk) / O(k) ⊗ O(jσ(1)) ⊗ · · · ⊗ O(jσ(k))

γ γ   O(j) / O(j), σ(jσ(1),...,jσ(k))

σ⊗σ−1 O(k) ⊗ O(j1) ⊗ · · · ⊗ O(jk) / O(k) ⊗ O(jσ(1)) ⊗ · · · ⊗ O(jσ(k))

γ γ   O(j) / O(j). σ(jσ(1),...,jσ(k))

Definition 1.1.2. (Coalgebra) Let O be an operad. An O-coalgebra is a chain complex C together with chain maps θ : O(j) ⊗ C → Cj satisfying the following conditions.

4 Pk cA1: (Associativity) Let s=1 js = j, then the following diagram commutes:

γ⊗id O(k) ⊗ O(j1) ⊗ · · · ⊗ O(jk) ⊗ C / O(j) ⊗ C

θ  id ⊗θ Cj O θk  O(j ) ⊗ · · · ⊗ O(j ) ⊗ Ck / O(j ) ⊗ C ⊗ · · · ⊗ O(j ) ⊗ C. 1 k shuffle 1 k cA2: (Unit) The following diagram commutes:

=∼ R ⊗ C / C : γ⊗id θ  O(1) ⊗ C.

cA3: (Equivariance) Let σ ∈ Σj, then the following diagram commutes:

σ⊗id O(j) ⊗ C / O(j) ⊗ C

θ θ   j j C σ / C .

A morphisms of O-coalgebras is a chain map commuting strictly with all the above structure. The category of O-coalgebras will be denoted by coAlgO. Definition 1.1.3. (Comodule) Let O be an operad and C an O-coalgebra. A C-comodule is a chain complex D together with chains maps

λ : O(j) ⊗ D → D ⊗ Cj−1 satisfying the following conditions.

5 Pk cM1: (Associativity) Let s=1 js = j, then the following diagram commutes:

γ⊗id O(k) ⊗ O(j1) ⊗ · · · ⊗ O(jk) ⊗ D / O(j) ⊗ M

θ  id ⊗λ D ⊗ Cj−1 O λ⊗θk−1  O(j ) ⊗ · · · ⊗ O(j ) ⊗ D ⊗ Ck−1 / O(j ) ⊗ D ⊗ · · · ⊗ O(j ) ⊗ C. 1 k shuffle 1 k cM2: (Unit) The following diagram commutes:

=∼ R ⊗ D / D : γ⊗id θ  O(1) ⊗ D.

cM3: (Equivariance) Let σ ∈ Σj−1 ⊂ Σj, then the following diagram com- mutes: σ⊗id O(j) ⊗ D / O(j) ⊗ D

θ θ   D ⊗ Cj−1 / D ⊗ Cj−1. id ⊗σ

A morphisms of C-comodules is a homeomorphism of abelian groups commuting strictly with all the above structure. The category of O-comodules O is enriched over Ab• and will be denoted by coModC .

Example 1.1.4. The operad A has A(j) = Z[Σj] with unit map equal to the identity and product maps dictated by the equivariance formulas. An A-coalgebra C is the same thing as a coassociative coalgebra. The operad product encodes all of the iterates and permutations of the coproduct of the coalgebra. A C-comodule D in the operadic sense is an C-bicomodule in the classical sense.

Definition 1.1.5. (E∞-operad [13]) An operad O is said to be an E∞ operad if it satisfies:

E1: (Unital) O(0) = Z.

6 E2: (Σ-free) Each Σj acts freely on O(j). E3: (Contractible) Each O(j) has the homology of a point.

1.2 The operad S

This section collects results related to an E∞-operad studied by several re- searchers, whose combinatorial nature and explicit coaction on normalized chains makes it suitable for the applications of this work. The coaction goes back to Steenrod construction in [40] of a chain ap- proximation to the diagonal inclusion of triangulated spaces. The definition of an operad for which this coaction give rise to a natural coalgebra structure on the normalized chains of simplicial sets, appears in the proof of Deligne’s conjecture by McClure-Smith [24] and is treated under the name “sequence operad” by the same authors in [25], where they present a filtration of it by En-suboperads. Work by Berger-Fresse in [4] uses the same operad with the name “surjection operad”. Jonathan Potts’ thesis [28] describes this op- erad with the name “step operad” and Jones-Adamaszek relate it to join operations of augmented symmetric simplicial sets in [1]. The definition of this E∞-operad, which will be denoted S to stand for Steenrod, sequence, surjection or step, will be presented below together with its filtration by En-suboperads and its natural coaction on the normalized chain complex of simplicial sets.

Chain complex of S. Let S(r)d be the free abelian group generated by all functions from {1, . . . , r + d} to {1, . . . , r} quotiented by the submodule generated by all non-surjective functions and the surjections u for which there exists i ∈ {1, . . . , r + d − 1} so that u(i) = u(i + 1). Set S(0)0 = Z and S(0)d = 0 for d > 0. The module S(r)d is free and any of the generators u : {1, . . . , r + d} → {1, . . . , r} will be identified with its ordered image (u(1), . . . , u(r + d)), which will be referred to as the coordinates of u. To study examples a diagrammatical representation of the surjections proves useful, as an illustration that generalizes one has that (1, 2, 1, 3, 2) is

7 represented by

.

Define ∂ : S(r)d → S(r)d−1 by

r+d X u u 7→ εi · (u(1),..., ud(i), . . . , u(r + d)) i=1

u u with εi a sign to be specified. In order to determine εi , separate the co- ordinates of u into disjoint sets each characterized by one of the following properties: a) The value of the coordinate equals the value of a coordinate to its right i.e. u(i) = u(i + j) for some positive j. b) The value of the coordinate is different from all coordinates to its right but equal to one on its left i.e. u(i) 6= u(i + j) and u(i) = u(i − k) for all j and some k positives. c) The value of the coordinate is different from all other coordinates i.e. u(i) 6= u(j) for any i.

Consider the set {u(i1), u(i2),... } of coordinates satisfying a) indexed so that 0 u j−1 j < j implies i < i 0 . Define for them ε = (−1) . For coordinates u(i) j j ij u u satisfying b) define ε = −ε with u(i ) = u(i) satisfying a) and u(i 0 ) 6= u(i) i ij j j 0 u for all j > j. Coordinates u(i) satisfying c) need no definition of εi since (u(1),..., ud(i), . . . , u(r + d)) = 0. For example if u = (1, 2, 1, 3, 2) then u(1) and u(2) satisfy a), u(3) and u(5) satisfy b) and u(4) satisfy c) so

∂ = − + .

8 Operadic composition of S. The maps

◦k : S(r)d ⊗ S(s)e → S(r + s − 1)d+e

−1 are defined as follows. Let u⊗v ∈ S(r)d ⊗S(s)e and let {i1, . . . , in} = u (k). For any splitting π of the coordinates of v into n subsequences


v(l0), . . . , v(l1) v(l1), . . . , v(l2) ··· v(ln−1), . . . , v(ln) (1.1)

π set the coordinates of a new surjection u ◦k v to be obtained from those of u by first replacing u(ij) by the j-th subsequence of v, adding k − 1 to those coordinates and then adding s − 1 to the coordinates u(i) which are greater than n. Define X π π u ◦k v = εk · (u ◦k v) π π −1 with εk a sign to be specified. In order to do so, notice that u (k) = {i1, . . . , in} induces a collection of subsequences



··· u(ij−1 + 1), . . . , u(ij − 1) u(ij) u(ij + 1), . . . , u(ij+1 − 1) u(ij+1) ··· (1.2) π and that in order to do the replacements in the definition of u ◦k v we can think of passing the subsequences from (1.1) across those from (1.2). The π sign εk will then be computed following the Koszul sign rule after defining the notion of degree for subsequences of coordinates. Let w be an arbitrary surjection and

··· w(mt−1 + 1), . . . , w(mt) w(mt − 1), . . . , w(mt+1) ··· be the partition of w into (consecutive and disjoint) subsequences determined by the coordinates w(mt) satisfying a). The degree of a general subsequence is then defined to be one less than the number of these subsequences that it overlaps with. For example, to compute u ◦2 v = (2, 1, 3, 2, 1) ◦2 (1, 2, 1) we first com- pute the degrees of the relevant subsequences, which requires the counting of overlaps with (2)(1)(3, 2, 1) for subsequences of u and with (1)(2, 1) for those of v. Placing the degree as a subindex one obtains

(1)0(1, 2, 1)1

(2)0(1, 3)1(2)0(1)0 ; (1, 2)1(2, 1)0

(1, 2, 1)1(1)0.

9 Therefore,

◦2 equals

+ − − .

Symmetric action on S. Define an action of Σr on S(r)d by σ · u = σ(u(1)), . . . , σ(u(r + d))
.

For example,

(123) =

Lemma 1.2.1. The structure defined above makes S = {S(r)•} into an E∞-operad (see Definition 1.1.5).

Proof. The action of Σr on S(r)• is free since it is free on the first coordinate of any surjection. The proof that S(r)• has the homology of a point is part (c) of Theorem 2.15 in [25].

Filtration of S by En-operads Consider a surjection u in S(r)• and a pair i < j ≤ r. Define uij to be the sequence obtained from the sequence of coordinates of u by removing all elements different from i and j. For example, if u = (2, 1, 3, 1, 2) then u12 = (2, 1, 1, 2), u13 = (1, 3, 1) and u23 = (2, 3, 2). To each such sequence assign the number of pairs of distinct consecutive coordinates and name it the change number. Using the previous example one sees that the change number of all uij is 2. For any u define its filtration weight to be the largest change number among all possible uij and define Sn to be the suboperad generated by all surjections whose filtration weight is less than or equal to n. The following appears as Theorem 3.5 in [25].

10 Lemma 1.2.2. The constructions above defines a filtration by suboperads

S1 ≤ S2 ≤ · · · ≤ S∞ = S

n with S an En-operad.

S-coalgebra structure on normalized chains. Let ∆ be the simplicial category as described in Definition A.17 and recall from Example A.22 the functor C• : ∆ → Ab• whose left Kan extension along the Yoneda embedding defines the normalized chains of simplicial sets. The purpose of this section is to construct a compatible collection of maps

⊗k S(k) ⊗ C•[0, . . . , n] → C•[0, . . . , n] , indexed by k, n ≥ 0, determining a functor represented as a dotted arrow in the following commutative diagram

coAlgS : forget  sSet / Ab• . C•

Let [i1, . . . , il] be the image of an order preserving function [0, . . . , l] → [0, . . . , n]. This function induces a chain map C•[0, . . . , l] → C•[0, . . . , n] and the image the top dimensional generator of C•[0, . . . , l] will be identified with the generator [i1, . . . , il] in C•[0, . . . , n] if ij 6= ij+1 for all j, being 0 otherwise. Let u ∈ S(r)d be a surjection and [0, . . . , n] ∈ ∆. Let π stand for a choice of (r + d − 1) elements of {0, . . . , n} satisfying

0 = n0 ≤ n1 ≤ · · · ≤ nr+d−1 ≤ nr+d = n and associate to this choice π a collection of generators of C•[0, . . . , n]  r+d [ni−1, . . . , ni] i=0 with consecutive vertices. Such generators will be referred to as intervals. For k ≤ r define Lπ(k) to be 0 in case there exists a pair of intervals [ni−1, . . . , ni] and [nj−1, . . . , nj] with u(i) = u(j) = k and a common vertex, or define Lπ(k) to be the generator in C•[0, . . . , n] whose set of vertices is the union of the vertices of all intervals [ni−1, . . . , ni] with u(i) = k.

11 ⊗r For every choice of π define an element in C•[0, . . . , n] by

uπ[0, . . . , n] = Lπ(1) ⊗ Lπ(2) ⊗ · · · ⊗ Lπ(r) and set X u[0, . . . , n] = εu,π · uπ[0, . . . , n] π with εu,π a sign to be specified. Example 1.2.3. If u = (1, 2), the value of u[0, 1, .., n] is equal to

P [0, 1, . . . , n] = i [0, . . . , i] ⊗ [i, . . . , n]

0 0 1 0 1 . . . n = + + ··· + 0 1 . . . n 1 . . . n n, with signs computed to be all positive in Example 1.2.4.

Signs of the S-coalgebra structure In order to specify the sign εu,π one distinguishes between two types of intervals. a) Internal intervals [ni−1, . . . , ni] satisfy u(i) = u(i+j) for some positive j. b) Final intervals [ni−1, . . . , ni] satisfy u(i) 6= u(i + j) for all positive j.

Define the degree of an interval [ni−1, . . . , ni] to be ni − ni−1 + 1 if it is internal or ni − ni−1 if it is final. Consider the permutation taking

(u(1), u(2), . . . , u(r + d)) 7→ (1,..., 1, 2,..., 2,...,r,...,r)

per and obtain, by Koszul’s rule, a sign εu,π from the induced permutation of the graded intervals
[0, . . . , n1], [n1, . . . , n2],..., [nr+d−1, . . . , nr+d] .

Consider all internal intervals [ni−1, . . . , ni] to be the indexing set of the P pos sum ni. Let this sum be the exponent of a sign εu,π and set

per pos εu,π = εu,π · εu,π.

12 Example 1.2.4. Let u = (1, 2) as in Example 1.2.3. For any [0, . . . , n] and pos any π, all signs εu,π are positive since every [ni−1, . . . , ni] is final, so εu,π = 1, per and εu,π = 1 because (1, 2) is already in the correct order.

Example 1.2.5. Let u = (..., 2, 1, 2) be one of the two generators of S(2)d and consider [0, . . . , d] ∈ ∆ of the same dimension as the degree of u. In order to compute the coaction of u on [0, . . . , d], thought of as one of the generators of C•[0, . . . , d], one notices that the only choice for π

0 = n0 ≤ n1 ≤ · · · ≤ nr+d−1 ≤ nr+d = d leading to a non-zero uπ[0, . . . , d] satisfies ni 6= ni+1 for all i = 1, . . . , d. Be- cause of the relation between the degree of the surjection and the dimension of the simplex, this choice is unique and given by ni = i−1 for all i = 1, . . . , d, in other words

0 ≤ 0 < 1 < 2 < ··· < (d − 1) < d ≤ d, so up to a sign εd one has   0 1 . . . d ··· [0, 1, . . . , d] = ε · d 0 1 . . . d

= εd · [0, 1, . . . , d] ⊗ [0, 1, . . . , d].

In order to determine the sign εd notice that only the last two intervals [d − 1, d] and [d, d] are final with degrees 1 and 0 respectively. All other intervals [i − 1, i] are internal and have degree 2 except for [0, 0] which has degree 1. Therefore, a permutation contributes with a negative sign if and only if it exchanges [0, 0] and [d − 1, d]. Consequently, the permutation

(..., 2, 1, 2) 7→ (1, 1,..., 2, 2)

per d pos induces the permutation sign εd = (−1) . The position sign εd , deter- Pd−1 mined by the internal intervals, is equal in this case to i=0 i so

d−1 per pos d X d(d+1)/2 εd = εd · εd = (−1) · i = (−1) . i=0

13 Example 1.2.6. Let u = (1, 2, 3, 1) and [0, 1, 2] ∈ ∆, this example will compute u[0, 1, 2]. A choice of

π : 0 = n0 ≤ n1 ≤ n2 ≤ n3 ≤ n4 = 2 leads to a non-zero term uπ if and only if n1 6= n3. The following table summarizes for such possible choices the degrees of the associated internal and final intervals, as well as the resulting permutation and position signs.

per pos n1 n2 n3 |[n0, . . . , n1]| |[n1, . . . , n2]| |[n2, . . . , n3]| |[n3, . . . , n4]| ε ε 0 0 1 1 (i) 0 (f) 1 (f) 1 (f) -1 1 0 0 2 1 (i) 0 (f) 2 (f) 0 (f) 1 1 0 1 1 1 (i) 1 (f) 0 (f) 1 (f) -1 1 0 1 2 1 (i) 1 (f) 1 (f) 0 (f) 1 1 0 2 2 1 (i) 2 (f) 0 (f) 0 (f) 1 1 1 1 2 2 (i) 0 (f) 1 (f) 0 (f) 1 -1 1 2 2 2 (i) 1 (f) 0 (f) 0 (f) 1 -1

Therefore,

[0, 1, 2] equals 0 1 2 0 2 0 1 2 − 0 + 0 − 0 1 0 1 0 1 2 1

0 2 0 2 0 1 2 0 1 2 + 0 1 + 0 1 2 − 1 − 1 2 1 2 2 1 2 0 2. Lemma 1.2.7. The maps define above

⊗k S(k) ⊗ C•[0, . . . , n] → C•[0, . . . , n]

14 determine a functor which is represented by the dotted arrow in the following commutative diagram

∆ / coAlgS

forget C • #  Ab• . Proof. This follows from part (b) of Theorem 2.15 in [25], see also part (b) of Remark 2.16 in the same reference. Definition 1.2.8. As in Definition A.19, a functor can be constructed by taking the left Kan extension along the Yoneda embedding of the functor of Lemma 1.2.7. Such functor is represented by the dotted arrow in the following commutative diagram

coAlgS : forget  sSet / Ab• C• and the image of any simplicial set X by this functor will be referred to as the S-coalgebra structure on C•(X). For any 1 ≤ k ≤ ∞, let Sk(2) denote the suboperad generated by the arity 2 part of the k-level of the filtration described in Lemma 1.2.2. Composing the above functor with the forgetful functor one has

coAlgS 9 C• forget  sSet / coAlgSk(2), C• and the image of any simplicial set X by this functor will be referred to as k the S (2)-coalgebra structure on C•(X). Notation 1.2.9. Let X be a simplicial set and consider the S-coalgebra structure on C•(X). For every surjection u ∈ S(k)• one has an abelian group homomorphism ⊗k C•(X) → C•(X) .

For u = (..., 2, 1, 2) one of the generators of S(2)d, the associated map will be denoted ∆d :C•(X) → C•(X) ⊗ C•(X).

15 1.3 Simplicial sets and S-coalgebras

In this section, the first of the two main technical results of this work is presented as Theorem 1.3.5. It follows from it that the category of based ordered simplicial complexes embeds into the category of coalgebras over the E∞-operad S, see Corollary 1.3.8. It also implies that the category of pointed small categories fully embeds into the category of coalgebras over 2 the E2-operad S , see Corollary 1.3.10. Similar results at the level of the homotopy category of simplicial sets have been obtained by Mandell [20], Smirnov [38], Smith [39] and others.

Lemma 1.3.1. Let X,Y ∈ sSet and σ ∈ Xn. If f :C•(X) → C•(Y ) is a homomorphism satisfying (f ⊗ f) ∆n σ = ∆n fσ, then either

fσ = 0 or fσ ∈ Yn.

Proof. Identifying non-degenerate simplices with their corresponding chains, Example 1.2.5 shows that for any n-dimensional simplex ρ one has ∆n ρ = εn P (−1) ρ ⊗ ρ. In particular, the condition on f implies that fσ = i ai τi for some τi ∈ Yn. Therefore,

(f ⊗ f) ∆n σ = ∆n fσ implies X X X ai τi ⊗ ai τi = ai τi ⊗ τi. i i i From which it follows that

2 ai aj = 0 if i 6= j and ai = ai. The only way these equations are satisfied is if all but possibly one of the coefficients ai are zero, with the possible exception being equal to 1. It follows that fσ = 0 or fσ = τi for some τi ∈ Yn. The next definition comes from [23]. Definition 1.3.2. The following are three properties that a simplicial set X might have. (A) X has property A if every face of a non-degenerate simplex of X is non-degenerate.

16 (B) X has property B if the n + 1 vertices of any non-degenerate n-simplex of X are distinct.

(C) X has property C if for any set of n + 1 distinct vertices of X, there is at most one non-degenerate n-simplex of X whose vertices are the elements of that set.

Definition 1.3.3. (Based simplicial sets) A simplicial set is said to based if it comes with a chosen vertex ∗.A based simplicial map between based simplicial sets is a simplicial map of the underlying simplicial sets preserving the base point. Denote the category of based simplicial sets by sSet∗ and let (−)+ : sSet → sSet∗ be the functor adding a disjoint base point. Notice that C•(X+, ∗) is isomorphic to C•(X) as S-coalgebras. Terminology 1.3.4. A functor F : C → C0 is said to be faithful, respec- tively full, if for every a, b ∈ C the function

HomC(a, b) → HomC0 (F a, F b) is injective, respectively surjective.

(n) Theorem 1.3.5. Let sSet∗ denote the full subcategory of n-dimensional k based simplicial sets as described in Definition A.17. Let S be the En- suboperad of S which is the k-th level of the filtration described in Lemma 1.2.2, see also Definition 1.2.8.

1. The functor C•(−, ∗) : sSet∗ → Ab• if faithful.

(1) 2. The functor C•(−, ∗) : sSet∗ → coAlgS2(2) if full.

(2) 3. The functor C•(−, ∗) : sSet∗ → coAlgS3(2) if full when restricted to simplicial sets satisfying property A.

(3) 4. The functor C•(−, ∗) : sSet∗ → coAlgS4(2) if full when restricted to simplicial sets satisfying property B.

(n) 5. The functor C•(−, ∗) : sSet∗ → coAlgSn+1(2) if full when restricted to simplicial sets satisfying properties B and C.

To improve the readability of this work, the proof of Theorem 1.3.5 will be postponed until after a few corollaries are drawn from it.

17 Definition 1.3.6. (Ordered simplicial complexes) An ordered simplicial complex X = (V,S) is a pair consisting of a partially ordered set V and a collection S of nonempty subsets of V with each s ∈ S inheriting a total order, such that

∀v ∈ V, [v] ∈ S and s0 ⊂ s ∈ S ⇒ s0 ∈ S.

A map of ordered simplicial complexes (V,S) → (V 0,S0) is an order preserving map F : V → V 0 such that

0 [v1, ..., vk] ∈ S ⇒ [F v1, ..., F vk] ∈ S .

This category is denoted SC and its based version is denoted SC∗. Remark 1.3.7. The functor sending an ordered simplicial complex to the simplicial set whose non-degenerate n-simplices correspond to subsets in S of cardinality n − 1, and whose degenerate simplices are freely generated; is a full and faithful functor whose essential image is the full subcategory of simplicial sets satisfying properties B and C. See [25] for more on this. The categories SC and SC∗ will be identified with their essential image. Corollary 1.3.8. The functor

C•(−, ∗) : SC∗ → coAlgS(2) is full and faithful. Proof. Notice that an S(2)-coalgebra map is in particular an Sk(2)-coalgebra map for every k > 0. By part 1 of Theorem 1.3.5, C•(−, ∗) : SC∗ → coAlgS(2) is faithful and by part 5 it is full. Remark 1.3.9. A consequences of this corollary is that the category of based simplicial complexes fully embeds into that of coalgebras over an E∞-operad.

Corollary 1.3.10. Let Cat∗ be the category of pointed small categories, N the nerve functor as described in Example A.23, and π(1) the functor project- ing to the 1-dimensional skeleton. The composition

N π(1) (1) C• Cat∗ / sSet∗ / sSet∗ / coAlgS2(2) is a full and faithful functor.

18 Proof. The composition π(1) ◦ N is a full and faithful functor, while the (1) functor C• : sSet∗ → coAlgS2(2) is faithful and full by parts 1 and 2 of Theorem 1.3.5.

Remark 1.3.11. A consequences of this corollary is that the category of pointed small categories fully embeds into that of coalgebras over an E2 operad.

Proof of Theorem 1.3.5 The proof of the five parts of Theorem 1.3.5 will be parcelized into three independent groups, beginning with part 5 followed by part 1 and finished with the remaining three parts. This choice is made to increase the readability of this work by presenting the less computational proof first, followed by the increasingly tedious case-by-case analysis involved in the other proofs. Proof of 5. Let X(n) = (V,S) and Y (n) = (V 0,S0) be n-dimensional based ordered simplicial complexes and consider an Sn+1(2)-coalgebra map

(n) (n) f :C•(X , ∗) → C•(Y , ∗).

Identifying simplices with their corresponding chains, Lemma 1.3.1 implies for any σ ∈ S that fσ = 0 or fσ ∈ S0. In particular, for vertices one has that f[v] = 0 or f[v] is a vertex of Y (n). Define a F : V → V 0 by ( fv if fv 6= 0, F v = ∗ if fv = 0 or v = ∗.

It needs to be shown that this is a morphism of based ordered simplicial complexes inducing f. This directly follows from establishing the next claims:

Claim 1. If f[v1, ..., vk] 6= 0 then f[v1, ..., vk] = [F v1, ..., F vk] satisfying that F vi < F vi+1 for all i.

Claim 2. If f[v1, ..., vk] = 0 then F vi = F vi+1 for some i or fvi = ∗ for all i. Proof of Claim 1. For vertices it holds trivially. Assume it holds for simplices of dimension (k − 1) and let f[v1, ..., vk] = [w1, . . . , wk]. Since f is a chain map one has

X i X i (−1) [F v0,..., Fd vi, . . . , F vk] = (−1) [w0,..., wbi, . . . , wk]

19 and the induction hypothesis proves the claim. Proof of Claim 2. For vertices it holds trivially. Assume it holds for simplices of dimension (k − 1). Since f is a chain map one has

X i (−1) f[v0,..., vbi, . . . , vk] = 0.

If f[v0,..., vbi, . . . , vk] = 0 for all i, then the induction hypothesis finishes the proof. If not, there must exist a pair i < j so that f[v0,..., vbi, . . . , vk] = f[v0,..., vbj, . . . , vk] 6= 0. By Claim 1. that implies

[F v0,..., Fd vi, . . . , F vk] = [F v0,..., Fd vj, . . . , F vk] so F vi = F vi+1. The next two groups of proofs rely on the standard identities, listed below, satisfied by the degeneracy and face maps of simplicial sets. These identities will be used without comment throughout the proofs.

i) didj = dj−1di if i < j,

ii) disj = sj−1di if i < j, iii) disj = id if i = j or i = j + 1,

iv) disj = sjdi−1 if i > j + 1,

v) sisj = sj+1si if i = j. Proof of 1. Let F,F 0 : X → Y be based simplicial maps inducing the same chain map f. Identifying non-degenerate simplices with their corresponding chains, for any simplex σ, if fσ 6= 0 then F σ = fσ = F 0σ. Since there are no 0 degenerate 0-dimensional simplices, F and F agree on X0. Assume for an 0 induction argument that F and F agree up to a certain skeleton Xk−1 and let σ ∈ Xk. If σ = siρ is degenerate then, using the induction hypothesis,

0 0 F σ = F siρ = siF ρ = siF ρ = F siρ = F σ.

The case fσ 6= 0 was already treated so assume σ is non-degenerate with 0 0 0 fσ = 0. There must exist i, j and ρ, ρ such that F σ = siρ and F σ = sjρ with this data satisfying one of the following possibilities:

20 0 0 0 0 a) If j = i then ρ = ρ since ρ = diF σ = F diσ = F diσ = diF σ = ρ . It 0 0 follows that F σ = siρ = siρ = F σ.

0 0 b) If j = i + 1 then ρ = sidiρ and ρ = ρ since ρ = diF σ = F diσ = 0 0 0 0 0 F diσ = diF σ = disi+1ρ = sidiρ and ρ = di+1F σ = F di+1σ = F di+1σ = 0 0 0 0 0 di+1F σ = ρ . It follows that F σ = siρ = sisidiρ = si+1sidiρ = si+1ρ = F 0σ.

0 0 c) If j = i + k with k > 1 then ρ = si+k−1di+1ρ and ρ = sidi+k−1ρ since 0 0 0 0 ρ = di+1F σ = F di+1σ = F di+1σ = di+1F σ = di+1si+kρ = si+k−1di+1ρ 0 0 0 0 and ρ = di+kF σ = F di+kσ = F di+kσ = di+kF σ = di+ksiρ = sidi+k−1ρ . 0 0 Applying di+1 to the ρ = sidi+k−1ρ gives di+1ρ = di+k−1ρ. It follows that 0 0 0 0 F σ = siρ = sisi+k−1di+1ρ = si+ksidi+1ρ = si+ksidi+k−1ρ = si+kρ = F σ.

Proof of 2, 3 and 4. Lemma 1.3.1 will be use throughout this proof without mention, as will be the identification of non-degenerate simplices with their corresponding chains. Given a morphism f between the appropriate coalge- bras, the following case-by-case procedure constructs a based simplicial map F with C•(F, ∗) = f :

σ ∈ X0: a) fσ 6= 0: set F σ = fσ. b) fσ = 0 or σ = ∗: set F σ = ∗.

σ ∈ X1: a) fσ 6= 0: set F σ = fσ.

Since (f ⊗ f) ∆0 σ = ∆0 fσ and ∆0 σ = d1σ ⊗ σ + σ ⊗ d0σ one has

fdjσ = djfσ for j = 0, 1.

·) If fdjσ 6= 0 then F djσ = fdjσ = djfσ = djF σ for j = 0, 1.

·) If fdjσ = 0 then djfσ = 0, therefore djF σ = ∗ = F djσ for j = 0, 1.

21 b) fσ = 0: set
F σ = s0F d0σ = s0F d1σ .

Since ∂fσ = f∂σ one has fd0σ = fd1σ.

·) d0F σ = d0s0F d0σ = F d0σ.

·) d1F σ = d1s0F d0σ = F d0σ = F d1σ.

σ ∈ X2: (assuming Y has Property A) a) fσ 6= 0: set F σ = fσ.

By Property A, djF σ is non-degenerate for all possible j. It follows that

djfσ 6= 0 for j = 0, 1, 2.

Since (f ⊗f) ∆0 σ = ∆0 fσ and ∆0 σ = d1d2σ ⊗σ +d2σ ⊗d0σ +σ ⊗d0d0σ, one has in particular that fd2σ ⊗ fd0σ = d2fσ ⊗ d0fσ. It follows that

fd2σ = d2fσ and fd0σ = d0fσ.

Since (f ⊗ f) ∆1 σ = ∆1 fσ and ∆1 σ = d1σ ⊗ σ − σ ⊗ d0σ − σ ⊗ d2σ, one has in particular that fd1σ = d1fσ.

Therefore, djF σ = djfσ = fdjσ = F djσ for j = 1, 2, 3. b) fσ = 0:

Since f∂σ = ∂fσ implies fd0σ − fd1σ + fd2σ = 0 one has the following possibilities:

i) fd0σ = fd1σ 6= 0 & fd2σ = 0: set

F σ = s0F d0σ.

·) d0F σ = d0s0F d0σ = F d0σ.

·) d1F σ = d1s0F d0σ = F d0σ = F d1σ.

·) d2F σ = d2s0F d0σ = s0d1F d0σ = s0F d1d0σ = s0F d1d0σ = s0F d0d2σ = F d2σ.

22 ii) fd0σ = 0 & fd1σ = fd2σ 6= 0: set

F σ = s1F d1σ

·) d0F σ = d0s1F d1σ = s0d0F d1σ = s0F d0d1σ = s0F d0d0σ = F d0σ.

·) d1F σ = d1s1F d1σ = F d1σ.

·) d2F σ = d2s1F d1σ = F d1σ = F d2σ.

iii) fd0σ = fd1σ = fd2σ = 0: set

F σ = s0F d0σ.

·) d0F σ = d0s0F d0σ = F d0σ.

·) d1F σ = d1s0F d0σ = F d0σ = s0F d0d0σ = s0F d0d1σ = F d1σ.

·) d2F σ = d2s0F d0σ = d2s0s0F d1d0σ = d2s1s0F d0d2σ = s0F d0d2σ = F d2σ.

σ ∈ X3: (assuming Y has Property B) a) fσ 6= 0:

By Property A, djF σ and djdiF σ are non-degenerate for all possible i, j. It follows that

djfσ 6= 0 and djdifσ 6= 0 for all possible i, j.

Since (f ⊗f) ∆0 σ = ∆0 fσ and ∆0 σ = d1d2d3σ ⊗σ +d2d3σ ⊗d0σ +d3σ ⊗ d0d0σ + σ ⊗ d0d0d0σ, one has in particular that

fd0σ = d0fσ and fd3σ = d3fσ.

Since (f ⊗ f) ∆2 σ = ∆2 fσ and ∆2 σ = −d1σ ⊗ σ − d3σ ⊗ σ − σ ⊗ d0σ − σ ⊗ d2σ, one has in particular that fd1σ + fd3σ = d1fσ + d3fσ and fd0σ + fd2σ = d0fσ + d2fσ. It follows that

fd1σ = d1fσ and fd2σ = d2fσ.

Therefore, djF σ = djfσ = fdjσ = F djσ for j = 0, 1, 2, 3.

23 b) fσ = 0:

The following observation will restrict the cases to be analyzed. For any of the three specific pairs (i, j) = (0, 2), (0, 3) or (1, 3) one has

fdiσ 6= 0 or fdjσ 6= 0 imply fdiσ 6= fdjσ.

·) Assume fd0σ = fd2σ 6= 0. By property B d1d1F d0σ 6= d1d0F d0σ. But d1d1F d0σ = F d1d1d0σ = F d1d0d2σ = d1d0F d2σ = d1d0F d0σ.A contradiction.

·) Assume fd0σ = fd3σ 6= 0. By property B d1d1F d0σ 6= d1d0F d0σ. But d1d1F d0σ = F d1d1d0σ = F d1d0d3σ = d1d0F d3σ = d1d0F d0σ.A contradiction.

·) Assume fd1σ = fd3σ 6= 0. By property B d1d0F d0σ 6= d0d0F d0σ. But d1d0F d0σ = F d1d0d0σ = F d0d0d3σ = d0d0F d3σ = d0d0F d0σ.A contradiction.

Since f∂σ = ∂fσ implies fd0σ − fd1σ + fd2σ − fd3σ = 0 one has the following possibilities:

i) fd0σ = fd1σ 6= 0 & fd2σ = fd3σ = 0: set

F σ = s0F d0σ.

The faces of F d0σ = F d1σ are non-degenerate by property A. In particular 0 6= fd0d1 = fd0d2 and 0 6= fd2d0 = fd0d3 implying that

F d2σ = s0F d0d2σ and F d3 = s0F d0d3σ.

·) d0F σ = d0s0F d0σ = F d0σ.

·) d1F σ = d1s0F d0σ = F d0σ = F d1σ.

·) d2F σ = d2s0F d0σ = s0d1F d0σ = s0F d1d0σ = s0F d0d2σ = F d2σ.

·) d3F σ = d3s0F d0σ = s0d2F d0σ = s0F d2d0σ = s0F d0d3σ = F d3σ.

ii) fd0σ = 0 & fd1σ = fd2σ 6= 0 & fd3σ = 0: set

F σ = s1F d1σ.

The faces of F d1σ = F d2σ are non-degenerate by property A. In particular 0 6= fd0d1 = fd0d0 and 0 6= fd2d2 = fd2d3 implying that

F d0σ = s0F d0d0σ and F d3 = s1F d1d3σ.

24 ·) d0F σ = d0s1F d1σ = s0d0F d1σ = s0F d0d1σ = s1F d0d0σ = F d0σ.

·) d1F σ = d1s1F d1σ = F d1σ.

·) d2F σ = d2s1F d1σ = F d1σ = F d2σ.

·) d3F σ = d3s1F d1σ = s1d2F d1σ = s1F d2d1σ = s1F d1d3σ = F d3σ. iii) fd0σ = fd1σ = 0 & fd2σ = fd3σ 6= 0: set

F σ = s2F d2σ.

The faces of F d2σ = F d3σ are non-degenerate by property A. In particular 0 6= fd0d3 = fd2d0 and 0 6= fd1d3 = fd2d1 implying that

F d0σ = s1F d1d0σ and F d1 = s1F d1d1σ.

·) d0F σ = d0s2F d2σ = s1d0F d2σ = s1F d0d2σ = s1F d1d0σ = F d0σ.

·) d1F σ = d1s2F d0σ = s1d1F d2σ = s1F d1d2σ = s1F d1d1σ = F d1σ.

·) d2F σ = d2s2F d2σ = F d2σ.

·) d3F σ = d3s2F d2σ = F d2σ = F d3σ. iv) fd0σ = fd1σ = fd2σ = fd3σ = 0.

If fd1d1σ = fd1d2σ 6= 0 then in order to determine the values of F d1σ and F d2σ one needs to know if fd0d1σ = 0 or not and if fd0d2σ = 0 or not. Since fd0d1σ = fd2d0σ and fd0d2σ = fd1d0σ this choice also determines F d0σ and shows one of the combinations is impossible, namely fd0d1σ 6= 0 and fd0d2σ = 0. Since fd2d1σ = fd1d3σ and fd2d2σ = fd2d3σ this choice also determines F d3σ.

Similarly, if fd1d1σ = fd1d2σ = 0 then all fdidjσ = 0. Therefore, one has the following possibilities: iv-a. fd0d1σ = fd1d1σ = fd0d2σ 6= 0 & fd1d3 = 0: set

F σ = s1s0F d1d1σ.

·) F d0σ = s0F d0d0σ = s0F d0d1σ = s0F d1d1σ = s0d0s0F d1d1σ = d0s1s0F d1d1σ = d0F σ.

·) F d1σ = s0F d0d1σ = s0F d1d1σ = d1s1s0F d1d1σ = d1F dσ.

·) F d2σ = s0F d0d2σ = s0F d1d1σ = d2s1s0F d1d1σ = d2F σ.

·) F d3σ = s0F d0d3σ = s0s0F d0d0d3σ = s1s0F d0d0d3σ = s1s0F d1d0d1σ = s1s0F d1d1d1σ = d3s1s0F d1d1σ = d3F σ.

25 iv-b. fd1d1σ = fd2d1σ = fd0d2σ = fd0d3σ = fd2d0σ 6= 0: set

F σ = s0s1F d1d1σ.

·) F d0σ = s1F d1d0σ = s1F d0d2σ = s1F d1d1σ = d0s0s1F d1d1σ = d0F σ.

·) F d1σ = s1F d1d1σ = d1s1s1F d1d1σ = d1F σ.

·) F d2σ = s0F d0d2σ = s0F d1d1σ = d2s2s0F d1d1σ = d2s0s1F d1d1σ = d2F σ.

·) F d3σ = s0F d0d3σ = s0F d1d1σ = d3s2s0F d1d1σ = d3s0s1F d1d1σ = d3F σ. iv-c. fd1d1σ = fd2d1σ = fd2d2σ = fd2d3σ 6= 0 & fd1d0 = 0: set

F σ = s1s1F d1d1σ.

·) fd0σ = s0F d0d0σ = s0s0F d0d0d0σ = s0s0F d0d1d1σ = d0s1s1F d1d1 = d0F σ.

·) F d1σ = s1F d1d1σ = d1s1s1F d1d1σ = d1F σ.

·) F d2σ = s1F d1d2σ = s1F d1d1σ = d2s1s1F d1d1 = d2F σ.

·) F d3σ = s1F d1d3σ = s1F d1d1σ = s1d2s1F d1d1 = d3s1s1F d1d1 = d3F σ. iv-d. fdjdiσ = 0: set F σ = s0s0s0F d0d0d0σ.

F diσ = s0s0F d0d0diσ = s0s0F d0d0d0σ = dis0s0s0F d0d0diσ = diF σ.

26 Chapter 2

Abelian sheaves and S-comodules

2.1 Sheaf theory of posets

This section is divided into three parts. The first part builds on a know equivalence of categories between partially ordered sets and T0-Alexandrov spaces. Each of these categories is equipped with a duality, opposite poset and topology of closed sets respectively, and Lemma 2.1.5 shows that the categorical equivalence is equivariant with respect to these dualities. The second part uses the connection between posets and Alexandrov spaces to relate the categorical definition of sheaves and cosheaves with the topological one, presenting the close connection that arises between sheaves and cosheaves over posets. The third part specializes to sheaves and cosheaves over posets with values in an abelian category and characterizes their projective objects, showing the existence of enough projectives under suitable conditions.

Alexandrov spaces and partially ordered sets

Definition 2.1.1. (Alexandrov Spaces) A topological space (X, τ) is said to be an Alexandrov space if an arbitrary intersection of open sets is an open set. This extra condition allows for the definition of a new topology on any Alexandrov space given by all closed sets of the original topology. This topology will be called the dual topology an denoted τ c.

27 The full subcategory of topological spaces given by Alexandrov spaces satisfying the T0 separation axiom (i.e. for any pair of points there is an open set containing one of them, but not both) will be denoted AT0. Notice that c (X, τ) ∈ AT0 if and only if (X, τ ) ∈ AT0. Definition 2.1.2. Let (P, ≤) be a poset. The set P can be made into a topological space in two ways. Define the topology τ≥ in P to be generated def by the subsets b≥ = {a : b ≥ a} for all b ∈ P , and the topology τ≤ to be def generated by the subsets b≤ = {c : b ≤ c} for all b ∈ P .

Let (X, τ) be a T0-Alexandrov space. The set X can be made into a poset def T in two ways. For any x ∈ X let Ux = x∈U∈τ U. Define X⊂ = (X, ≤) with x ≤ y if and only if Ux ⊂ Uy, and define X⊃ = (X, ≤) with x ≤ y if and only if Ux ⊃ Uy. Lemma 2.1.3. The four assignments described above are functorial. More- over, they define pairs of inverse functors

(−)≥ : Poset  AT0 :(−)⊂ and (−)≤ : Poset  AT0 :(−)⊃. Proof. Given an order preserving function f : P → P 0 one needs to prove that 0 f is continuous with respect to both topologies. Let c≥ be a basis element of 0 −1 0 τ≥ and consider f (c≥). This set is open since it is straightforward to check that −1 0 [ f (c≥) = b≥. {b: c0≥f(b)} 0 0 Analogously, for a basis element a≤ of τ≤ one has

−1 0 [ f (a≤) = b≤. {b: a0≤f(b)} Given a continuous function f : X → X0 one needs to prove that if −1 Ux ⊂ Uy then Uf(x) ⊂ Uf(y). To do so, notice that f (Uf(y)) is an open set containing y and, since Uy is the smallest open set with that property, Uy ⊂ −1 f (Uf(y)). The assumption Ux ⊂ Uy together with the previous observation imply that f(x) ∈ f(Ux) ⊂ f(Uy) ⊂ Uf(y),

28 which, since Uf(x) is the smallest open set containing f(x), give the desired Uf(x) ⊂ Uf(y). Verifying these pairs of functors are inverse of each other follows directly from noticing that in τ≥ one has Ub = b≥, while in τ≤ one has Ub = b≤.

Definition 2.1.4. The (covariant) functor associating to a T0-Alexandrov space the T0-Alexandrov space with the dual topology is denoted by c (−) : AT0 → AT0 . The (covariant) functor associating to a poset the poset with the opposite order is denoted by (−)op : Poset → Poset . Lemma 2.1.5. The functors defined in this section are related by the follow- ing identities: op op 1a. (−)≥ = (−)≤ ◦ (−) . 1b. (−)≤ = (−)≥ ◦ (−) .

c c 2a. (−)≥ = (−) ◦ (−)≤. 2b. (−)≤ = (−) ◦ (−)≥.

op op 3a. (−)⊂ = (−) ◦ (−)⊃. 3b. (−)⊃ = (−) ◦ (−)⊂.

c c 4a. (−)⊂ = (−)⊃ ◦ (−) . 4b. (−)⊃ = (−)⊂ ◦ (−) . Proof. Pairs of corresponding identities are equivalent since (−)op ◦ (−)op = id and (−)c ◦ (−)c = id . The third pair of identities follows from the first pair since

(−)≥ ◦ (−)⊂ = id and (−)⊃ ◦ (−)≤ = id . The fourth pair of identities follows from the second pair since

(−)⊂ ◦ (−)≥ = id and (−)≤ ◦ (−)⊃ = id .

Proof of 1a. For any poset (P, ≤) the topology τ≥ is generated by sets {a : op b ≥ a}, while the topology τ≤op is generated by sets {a : b ≤ a}. These bases are equal so τ≤ = τ≥op . Proof of 2a. Consider an arbitrary poset (P, ≤). It will be shown first that c τ≥ ⊂ τ≤. To do so, consider a generator {a : b ≥ a} of τ≥ and notice that  c [ c {a : b ≥ a} ⊂  {y : x ≤ y} ∈ τ≤ {x : bx}

29 since b ≥ a and a ≥ x implies b ≥ x. Also, one has [ {a : b ≥ a}c ⊂ {y : x ≤ y} {x : bx} since x ∈ {a : b ≥ a}c implies x ∈ {y : x ≤ y} with b  x. c c In order to show that τ≥ ⊃ τ≤ consider an arbitrary open set of τ≤ say S
c T c x∈I {y : x ≤ y} = x∈I {y : x ≤ y} . Since τ≥ is closed under arbitrary c intersections, it suffices to show that {y : x ≤ y} is open in τ≥. This follows from a computation similar to the one above showing that

c [ {y : x ≤ y} = {a : b ≥ a} ∈ τ≥, {b : xb} and concludes the proof.

Sheaves, cosheaves and their relationship over posets Definition 2.1.6. (Presheaves and precosheaves) Let C and V be categories. A presheaf, respectively precosheaf, on C with values on V is a contravari- ant, respectively covariant, functor from C to V.A presheaf morphism, respectively precosheaf morphism, is a natural transformation of such functors. Denote these categories respectively by PSh(C, V) and PcoSh(C, V), or if V is understood from the context, simply by PSh(C) and PcoSh(C). Definition 2.1.7. (Sites) A site is given by a (small) category C and a set Cov(C) of families of morphisms with fixed target {Ui → U}i∈I , called coverings of C, satisfying the following axioms: S1: If V → U is an isomorphism then {V → U} ∈ Cov(C).

S2: If {Ui → U}i∈I ∈ Cov(C) and for each i ∈ I one has that {Vij →

Ui}j∈Ji ∈ Cov(C), then {Vij → U}i∈I,j∈Ji ∈ Cov(C).

S3: If {Ui → U}i∈I ∈ Cov(C) and V → U is a morphism in C then the pullback Ui ×U V exists for each i ∈ I and {Ui ×U V → V }i∈I ∈ Cov(C). Example 2.1.8. Let (X, τ) be a topological space. Think of τ as a category with set of objects τ and ( {V → U} if V ⊂ U, Homτ (V,U) = ∅ if V 6⊂ U,

30 and notice that this assignment Top → Cat is functorial. Define the set of coverings of τ by [ {Ui → U}i∈I ∈ Cov(τ) if and only if Ui = U. i∈I The conditions for τ with this coverings to define a site are easily verified.

Remark 2.1.9. The functors (−)≥ and (−)≤ from Lemma 2.1.3 can be com- posed with the functor described in the previous example. Abusing notation, these resulting functors are denoted (−) : Poset → Cat (−) : Poset → Cat ≥ & ≥ (P, ≤) 7→ τ≥. (P, ≤) 7→ τ≥. Example 2.1.10. Given any (small) category, define a site by declaring the coverings to be the identity morphisms only. In particular, since the assignment that takes any poset (P, ≤) to a category with set of objects P and morphisms ( {x → y} if x ≤ y HomP (x, y) = ∅ if x  y is a full and faithful functor, any poset P can be thought of a site with trivial coverings. The following definitions use some of the examples of limits and colimits described in Appendix A. Definition 2.1.11. (Sheaves and cosheaves) Let C be a site, D ∈ PSh(C) and E ∈ PcoSh(C). The presheaf D is said to be a sheaf if for all {Ui → U}i∈I ∈ Cov(C) the first arrow in the following diagram represents the equalizer of the next two Y Y D(U) → D(Ui) ⇒ D(Ui ×U Uj). i∈I i,j∈I The full subcategory of sheaves in PSh(C) will be denoted by Sh(C). The precosheaf E is said to be a cosheaf if for all {Ui → U}i∈I ∈ Cov(C) the last arrow in the following diagram represents the coequalizer of the first two a a E(Ui ×U Uj) ⇒ E(Ui) → E(U). i,j∈I i∈I The full subcategory of cosheaves in PcoSh(C) will be denoted by coSh(C).

31 Example 2.1.12. Let (X, τ) be a topological space. Sheaves and cosheaves on the site τ, defined in Example 2.1.8, agree with the usual topological sheaves and cosheaves on X. Example 2.1.13. If a site has coverings given by the identity morphisms only, then the categories of sheaves and presheaves on such site agree; as also do the categories of cosheaves and precosheaves on it. In particular, following Example 2.1.10, for any poset (P, ≤) one has Sh(P ) = PSh(P ) and coSh(P ) = PcoSh(P ) over this indiscrete site. Definition 2.1.14. Let (P, ≤) be a poset and V a complete and cocomplete category. From Remark 2.1.9 one obtains a covariant functor (−)≥ : P → τ≥ when P is regarded as a category. Define the functor

Lan : Sh(P, V) → PSh(τ≤, V),

op which assigns to any D ∈ Sh(P, V) the left Kan extension of D along (−)≥ , diagrammatically D P op / V ; (−)op ≥ Lan D  op (τ≥) . op The functor Ran : coSh(P , V) → PcoSh(τ≤, V) is defined using the con- travariant functor (−)≤ : P → τ≤ in a similar manner, diagrammatically

E P op / V < (−)≤ Ran E  τ≤.

Lemma 2.1.15. Let (P, ≤) be a poset and V a complete and cocomplete category. For any D ∈ Sh(P, V) the presheaf Lan D is a sheaf and the functor

Lan : Sh(P, V) → Sh(τ≥, V) is an equivalence of categories. Similarly, for any E ∈ coSh(P op, V) the precosheaf Ran E is a cosheaf and the functor

op Ran : coSh(P , V) → coSh(τ≤, V) is an equivalence of categories.

32 Proof. One needs to verify that the presheaf Lan D satisfies the sheaf condi- tion. For any U ∈ τ≥, Lemma A.16 and Lemma A.7 provide a formula for the left Kan extension Y Y
Lan D(U) = eq D(y) ⇒ D(x) , y≥⊂ U x≥⊂ y≥ which is exactly the sheaf condition for the finest cover of U. (Recall that the collection {y≥ : y ∈ P } forms a basis of τ≥). The inverse functor of Lan is given by restricting a sheaf on τ≥ to the basis {y≥ : y ∈ P }. The proof for cosheaves is analogous using Lemma A.16 and Lemma A.12.

Definition 2.1.16. Let (P, ≤) be a poset. Consider the contravariant func- c c c tor (−) : τ≥ → (τ≥) taking U to U and recall from Lemma 2.1.5 that c τ≥ = τ≤. Abusing notation, define the functor

c (−) : Sh(τ≥) → PcoSh(τ≤)

c which assigns to every D ∈ Sh(τ≥) the precosheaf D ∈ PcoSh(τ≤) defined by the following commutative diagram

op D (τ≥) / V O O (−)c Dc

c = τ≥ o τ≤ .

Lemma 2.1.17. Let (P, ≤) be a poset and V a complete and cocomplete cat- egory where the sheaves and cosheaves under consideration take their values. c For any D ∈ Sh(τ≥) the precosheaf D is a cosheaf and the functor

c (−) : Sh(τ≥) → coSh(τ≤) is an equivalence of categories making the following diagram commute

= Sh(P ) / coSh(P op)

Lan Ran   Sh(τ≥) / coSh(τ≤) (−)c

33 Proof. It suffices to establish the commutativity of the diagram since Lan and Ran are equivalence of categories. To do so, consider the following diagram associated to any D ∈ Sh(P )

op (−)≥ P op

D opÑ Lan D | (τ≥) / V O F O

(−)≤ (−)c (Lan D)c Ran D

c y τ≥ o = / τ≤.

c c Since (Lan D) ◦ (−)≥ = Lan D ◦(−) ◦ (−)≤ = Lan D ◦(−)≥ = D, the uni- versal property of right Kan extensions ensures the existence of a natural transformation (Lan D)c → Ran D. The inverse natural transformation is obtained similarly using the universal property of left Kan extensions and c the functor (−) : Sh(τ≥) → coSh(τ≤).

Abelian sheaves over posets Definition 2.1.18. (Projective objects) Let A be an abelian category. An object P ∈ A is called projective if it satisfies any of the following equivalent conditions:

1. For any surjection f : C → B and any map q : P → B there exists g : P → C such that q ◦ g = f, diagrammatically C ? g q  / B P f

 0. 2. Any exact sequence

0 → A → B → P → 0

34 splits, i.e. it is isomorphic to

0 → A → A ⊕ P → P → 0 with inclusion and projection maps.

Remark 2.1.19. The dual notion of an injective object will be omitted since it is not used in this work.

Definition 2.1.20. (Elementary projective sheaves and cosheaves) Let P ∈ A be a projective object, (P, ≤) a poset and y an element in P . The elementary projective sheaf P≤y ∈ Sh(P, Ab) with value P over y is defined by ( P if x ≤ y P≤y[x] = 0 if x  y, with all non-zero morphisms equal to the identity. The elementary projective cosheaf Py≤ ∈ coSh(P, Ab) with value P over y is defined by ( P if y ≤ z Py≤[z] = 0 if y  z, with all non-zero morphisms equal to the identity.

Terminology 2.1.21. A poset (P, ≤) is said to be locally finite if for all pairs x, z ∈ P the set {y ∈ P : x ≤ y ≤ z} is finite.

Lemma 2.1.22. Let A be a cocomplete abelian category and (P, ≤) a poset. A sheaf or a cosheaf over P with values in A is projective if, and if P is locally finite, only if, it is isomorphic to a direct sum of elementary projective ones. Proof. Only the proof for sheaves will be presented since small variations adapt it for cosheaves. Notice that by the universal property of coprod- ucts a direct sum of projective objects is projective. Explicitly, a morphism L Pi → B defines a collection of morphism Pi → B by precomposing with the respective inclusion. given a surjection A → B one gets a collection of L lifts Pi → A and therefore a lift Pi → A. A sheaf P≤y or cosheaf Py≤ is projective since it is straightforward to check that HomSh(P≤y, D) = Hom(P≤y[y], D[y])

35 and HomcoSh(Py≤, E) = Hom(Py≤[y], E[y]). Let P be a projective sheaf and y an element of the poset. Notice that by considering sheaves with support only on y one concludes that P[y] is a projective object in A. By the local finiteness of the poset under consider- ation every set {z : z > y} as a smallest element. Define the near star of y, denoted nst(y), as the sub-poset containing all such minimal elements. Notice that all pairs of elements in this sub-poset are not comparable. Define the sheaf Qy by ( P[z] if z ∈ nst(y) Qy[z] = 0 if z 6∈ nst(y). ˜ with all induced morphisms the zero map. The sheaf Qy is defined by   Py[z] if z ∈ nst(y)  ˜  0 if z 6∈ nst(y) and z 6= y Qy[z] = M  P[z0] if z = y  z0∈ nst(y) with non zero induced morphisms given by the inclusions M P[z] → P[z0]. z0∈ nst(y) There are obvious surjections represented by solid arrows below ˜ Qy g ?

 P / Qy

 0 and the morphism g[z] is the identity for all z ∈ nst(y) and it is zero for all z 6∈ nst(y) such that z 6= y. The collection of maps P[z] → P[y] for z ∈ nst(y) L gives a map z∈nst(y) P[z] → P[y] fitting into the following sequence M g[y] M 0 −→ P[z] −→ P[y] −→ P[z] −→ 0 z∈nst(y) z∈nst(y)

36 L which is exact since, denoting the inclusion P[z] → z∈nst(y) P[z] by ιz, for every z > y one has

g[y] ◦ P[y < z] = ιz ◦ g[z] = ιz. The above sequence splits by the projectivity of the objects involved, see Definition 2.1.18, so there exists B(y) such that M P[y] ∼= B(y) ⊕ P[z]
z∈nst(y) iterating the argument one gets M P[y] ∼= B(z) z≥y for every y and therefore M P = B(y)≤y y∈P concluding the proof. Remark 2.1.23. The condition in Lemma 2.1.22 that A be cocomplete is in practice too restrictive. For example, the category Abf of finitely generated abelian groups is not cocomplete. The conclusion of Lemma 2.1.22 remains true if one restricts to appropriate subcategories of sheaves or cosheaves on a poset where the relevant coproducts exist. Examples of such subcategories are the following. Definition 2.1.24. (Sheaves and cosheaves with compact support) Let P be a poset and A an abelian category. Denote by Shc(X, A) the full subcategory of Sh(X, A) whose objects satisfy D[x] 6= 0 for at most finitely many x ∈ P . Define coShc(X, A) similarly. Definition 2.1.25. (Enough projectives) An abelian category is said to have enough projectives if for any object B ∈ A there exists a surjection A → B → 0 with A ∈ A projective. Lemma 2.1.26. Let P be a poset and A an abelian category. If A has enough projectives then Shc(P, A) and coShc(P, A) do so as well. If in addition A is cocomplete, then Sh(P, A) and coSh(P, A) also have enough projectives.

37 Proof. Assume A is cocomplete and let D ∈ Sh(X, A). The first step is to construct a surjection of sheaves Q → D → 0 with Q[x] ∈ A projective for every x ∈ P . Choose for each x ∈ P a surjection f[x]: Q[x] → D[x] → 0 with Q[x] projective. For every pair x ≤ y there is a morphism D[y] → D[x] whose precomposition with f[y]: Q[y] → D[y] is represented by the horizontal solid arrow in the following diagram Q[x] < f[x]  Q[y] / D[x] < f[y]   D[y] 0. The choice of a morphism realizing the dotted arrow for any pair x ≤ y makes Q into a sheaf and f into a surjective morphism of sheaves. Define P ∈ Sh(X) by M P[x] = Q[y]≤y. x≤y This sheaf maps surjectively onto X, therefore onto D, and it is projective by Lemma 2.1.22. All other statements are proven in the same manner. + Notation 2.1.27. Let A be an abelian category. Denote by A• the cat- egory of bounded below complexes whose objects are homologically + graded complexes which are zero below some degree. Notice that A• is enriched over Ab•, and as usual one says that two morphisms are chain homotopy equivalent if they are homologous. The following are standard result in homological algebra, see for example [45] section 5.7 for their proofs. Lemma 2.1.28. Let A be an abelian category with enough projectives. + + 1. For any A• ∈ A• there exists a complex of projective objects P• ∈ A• and a morphism P• → A• inducing an isomorphism in homology.

0 0 2. If P• and P• are projective and f : P• → P• induces an isomorphism in 0 homology, then there exists g ∈ Hom + (P , P ) ∈ Ab such that f ◦ g A• • • • and g ◦ f are chain homotopy equivalent to the respective identities.

38 2.2 Simplicial complexes and Ranicki duality

In this section, the theory developed in the previous one is specialized in two ways. Only posets associated with simplicial complexes are considered and the sheaves and cosheaves studied have values either in the category of abelian groups Ab or its full subcategory Abf of finitely generated abelian groups. The notion of tensor product of functors is used to define the tensor product of a complex of sheaves and a complex of cosheaves. In conjunction with linear duality and a couple of special complexes, this tensor product is used to define the Ranicki duality functors, whose geometric content is made apparent by the pair subdivision sheaf and cosheaf. The section closes with the construction, using the pair subdivision sheaf, of the visible symmetric complex of a regular pseudomanifold. Definition 2.2.1. Consider the category SC of ordered simplicial complexes as presented in Definition 1.3.6. Define a functor

SC → Poset sending an ordered simplicial complex X = (V,S) to the poset (S, ≤) with σ ≤ τ if an only if σ ⊂ τ, i.e. if σ is a face of τ. For any X ∈ SC, define the categories of sheaves and cosheaves on X, denoted Sh(X) and coSh(X), to be the corresponding sheaves and cosheaves categories on its associated poset. The barycentric subdivision functor SC → SC is defined as the com- position of the functor defined above and the functor Poset → SC sending a poset (P, ≤) to the simplicial complex with P as set of vertices and simplices given by strictly ascending sets [σ0 < ··· < σn] of elements in P . Definition 2.2.2. (Open star and closure) Let X be an ordered simplicial complex and σ a simplex in X. The open star of σ, denote by st σ, is defined as the subset of X formed by all simplices containing σ. The closure of σ, denote by cl σ, is defined as the subcomplex of X formed by all simplices contained in σ. Notice that if σ ≤ τ then st σ ⊃ st τ and cl σ ⊂ cl τ. Definition 2.2.3. The complex of sheaves C• with values in Ab is defined to assign to each simplex the chain complex of cochains on its open star, i.e.

C•[σ] = C•(st σ), δ
,

39 and to have induced morphisms given by inclusions. The complex of cosheaves C• with values in Ab is defined to assign to each simplex the chain complex of cochains on its closure, i.e.
C•[σ] = C•(cl σ), ∂ , and to have induced morphisms given by inclusions. The following definition is presented in level of generality suitable for the purposes of this work. For a more general discussion see for example [35]. Definition 2.2.4. (Tensor products of functors) Consider a pair of functors F : Cop → Ab and G : C → Ab. The tensor product of F and G over C is defined by  M M  F ⊗ G = coeq F (c2) ⊗ G(c1) ⇒ F (c) ⊗ G(c) , C f: c1→c2 c and the tensor product of G and F over C is defined by  M M  G ⊗ F = coeq G(c1) ⊗ F (c2) ⇒ F (c) ⊗ G(c) . C f: c1→c2 c Example 2.2.5. Let R be a ring thought of as a category enriched over Ab with a single object. A right R-module corresponds to an Ab-enriched functor A : Rop → Ab, while a left R-module corresponds to an Ab-enriched functor B : R → Ab. The functor tensor product A ⊗R B agrees with the usual tensor product of a left and a right R-module. Definition 2.2.6. Given D ∈ Sh(X) and E ∈ coSh(X) define the sheaf E D by st M  (E  D)[σ] = E |st σ ⊗ D |st σ = E[τ] ⊗ D[τ] ∼ st st σ σ≤τ with D[ι](e) ⊗ d
∼ e ⊗ E[ι](d)
for any ι : σ → τ, and morphisms being induced by inclusions. Analogously, define the cosheaf E D cl M (E  D)[σ] = E |cl σ ⊗ D |cl σ = E[ρ] ⊗ D[ρ]/ ∼ cl cl σ ρ≤σ

40 with E[ι](e) ⊗ d
∼ e ⊗ D[ι](d)
for any ι : ρ → σ, and morphisms being induced by inclusions. These assignments are functorial in both variables. The notation st and cl will also be used for the extension of these bifunctors to complexes. Remark 2.2.7. Fix an ordered simplicial complex X. Let Z be the constant cosheaf on X with value Z. For any sheaf D the collection of maps
 M  D [σ] = τ ⊗ D[τ]/ ∼ → D[σ] Z st Z σ≤τ sending 1τ ⊗ d to D[ι](d) with ι : σ → τ defines an isomorphism of sheaves. Analogously, let Z be the constant sheaf. For any cosheaf E the collection of maps
 M  E Z [σ] = E[ρ] ⊗ Zρ/ ∼ → E[σ] cl ρ≤σ sending e ⊗ 1ρ to E[ι](e) with ι : ρ → σ defines an isomorphism of cosheaves. Notation 2.2.8. Let (−)∨ : Sh(X, Ab) → coSh(X, Ab) be the functor in- duced from linear duality. Explicitly, for any D ∈ Sh(X, Ab) one has

D∨[σ] = (D[σ])∨ and D∨[ι] = (D[ι])∨ for all ι : σ → τ.

Since the context will be clear enough to avoid confusions, the analogous functor coSh(X, Ab) → Sh(X, Ab) will be denoted by the same symbol (−)∨. The same notation (−)∨ will be used for the extension of these functors to complexes.

Definition 2.2.9. (Ranicki duality functors) Let X be an ordered simplicial complex. Abusing notation, define respectively the Ranicki duality func- tors T : Sh(X, Ab)• → Sh(X, Ab)• and T : coSh(X, Ab)• → coSh(X, Ab)• as the following compositions

∨ • ∨ T(−) = (−) C and T(−) = C• (−) . st cl

∨ ∼ ∨ ∼ Example 2.2.10. Notice that Z = Z and Z = Z, so by Remark 2.2.7 one ∼ • ∼ has TZ = C and TZ = C•.

41 • Example 2.2.11. Let X be the interval an represent C and C• by

C• : α γ C• : −1 b +1 −1 +1       βo ? _ β / β a / a co ? _ c • • • •.

Their linear duals are represented by

• ∨ ∨ (C ) : b / / bo o b (C•) : αo o α γ / / γ −1 +1 −1  Ñ +1 a c β • • • •.

The Ranicki dual of C• defined as (C•)∨ C• is represented by st TC• : bα bγ −1 +1 −1 +1 }  ~ aα bβo ? _ bβ / bβ cγ

• •.

∨ The Ranicki dual of C• defined as C• (C•) is represented by cl

TC• : bα bγ −1 +1 −1 +1  } ~ aα / aα bβ cγo ? _ cγ

• •. Observe that in this example the collection of evaluation maps

• ∨ • (C [σ]) ⊗ C [σ] → Z induces a morphism of complexes of sheaves

• 2 ε :TC = T Z → Z with ε[σ] a homology isomorphism for every simplex σ.

42 Similarly defined, the morphism

2 ε :TC• = T Z → Z is a homology isomorphism over each simplex. Definition 2.2.12. (Pair subdivision) For any finite ordered simplicial com- Sh plex define its pair subdivision sheaf P• as the Ranicki dual of the com- plex of sheaves C•. Explicitly, one has chain isomorphisms

Sh ∼ M ∗
M ∗
P• [ρ] = τ ⊗ σ / τ ⊗ σ ρ≤σ,τ στ

Sh 0 Sh 0 Sh with maps P• [ρ → ρ ]:P• [ρ ] → P• [ρ] given by inclusions. Define the pair subdivision cosheaf similarly by

cSh ∨ P• = T C• = C• (C•) . cl Remark 2.2.13. The pair subdivision sheaf has a geometric interpretation justifying its name. Let X be an ordered simplicial complex and X0 its barycentric subdivision as defined in 2.2.1. For any ρ ∈ X define the close dual cone of ρ, denoted dc(ρ), as the subcomplex of X0 containing all sim- Sh plices of the form [ρ1 < ρ2 < ··· ] with ρ ≤ ρ1. The chain complex P• [ρ] is chain isomorphic to the chain complex of a regular CW complex obtained by gluing along common faces certain simplices of dc(ρ). Two simplices in the closed dual cone are amalgamated along their common face if they have the same dimension and are represented by ascending subsets with the same initial and terminal simplices. For example, in the case of the (geometric realization of the) 2-dimensional simplex the amalgamation map looks as follows:

−→

The subdivision map, which is the inverse of the amalgamation one de- Sh fined above, induces a chain homotopy equivalence from P• [ρ] into the chains on the dual cone of ρ which are denoted D•[ρ]. Notice that D• defines a com- Sh plex of projective sheaves and, since P• is also projective, the complexes Sh D• and P• are chain homotopy equivalent be Lemma 2.1.28. In particular, Sh P• [ρ] is contractible for all ρ ∈ X. See [36] for more on the pair subdivision complex.

43 The Ranicki duality functors T are not in general involutions, not even up to homotopy. In order to have an involution-like property one needs to impose some finiteness conditions. This is accomplished by restricting T to Shc(X, Abf )• and coShc(X, Abf )•, i.e. the categories of complexes of com- pactly supported sheaves, respectively cosheaves, with values in the category of finitely generated abelian groups. Lemma 2.2.14. There exist natural transformations

2 2 ε :T → idShc(X,Abf ) and ε :T → idcoShc(X,Abf ) defined below, such that if P• stands for a complex of projective sheaves or cosheaves then 1. The following pairs of complexes are chain isomorphic 2 ∼ Sh 2 ∼ cSh T P• = P• P• and T P• = P• P• st cl

2 2. The morphism εP• : (T P•) → P• is a chain homotopy equivalence. 3. The following diagram commutes

T(εP• ) 3 TP• / T P•

εTP• id #  TP•.

Proof. Only the proof for sheaves will be presented since small variations adapt it for cosheaves. For any simplex σ ∈ X denote its i-th face by ∂iσ i i i and by δ σ any simplex such that ∂iδ σ = σ, notice that if δ σ exists then it is unique. M ∨ •
For any complex of sheaf D• one has T D•[ρ] = D• [σ] ⊗ C [σ]/ ∼ ρ≤σ M ∨ • 0 0∗
which equals D• [σ] ⊗ C [σ ≤ σ ]σ / ∼ . Using that ρ≤σ≤σ0

∗ • 0 0∗ ∨ 0 ∗ 0∗ dσ ⊗ C [σ ≤ σ ](σ ) ∼ D• [σ ≤ σ ](dσ) ⊗ σ one sees that as graded abelian groups

∼ M ∨ ∗ T D•[ρ] = D• [σ] ⊗ σ . ρ≤σ

44 This isomorphism can be improved to a chain isomorphism by setting

∗ ∗ |dσ| X i ∨ i i ∗ ∂(dσ ⊗ σ ) = ∂dσ ⊗ σ + (−1) (−1) D• [σ ≤ δ σ](dσ) ⊗ (δ σ) . δiσ Applying the previous observation twice and using the finite dimension- ality of the abelian groups involved one has that as graded abelian groups 2 ∼ M ∗ T D•[ρ] = D•[τ] ⊗ τ ⊗ σ . ρ≤σ≤τ To describe the boundary making this into a chain isomorphism observe that ∨ 0 L L (T D•) [σ ≤ σ ]: σ≤τ D•[τ] ⊗ τ → σ0≤τ D•[τ] ⊗ τ is giving by projection. M ∗ 2 The boundary in D•[τ] ⊗ τ ⊗ σ making it chain isomorphic to T D•[ρ] ρ≤σ≤τ is therefore ∗ ∗ ∂(dτ ⊗ τ ⊗ σ ) = ∂(dτ ) ⊗ τ ⊗ σ

|dτ | X i ∗ + (−1) (−1) D•[∂iτ ≤ τ](dτ ) ⊗ ∂iτ ⊗ σ ∂iτ

|dτ |+|τ| X i i ∗ + (−1) (−1) dτ ⊗ τ ⊗ (δ σ) . δiσ≤τ

The above formula conceptually simplifies if D• is projective. In that case 2 D[∂iτ ≤ τ] is an inclusion map and the chain complex T [ρ] becomes chain Sh isomorphic to D[ρ] ⊗ P• [ρ]. 2 ∗ For any D• define εD• [ρ]:T D•[ρ] → D•[ρ] by sending dτ ⊗ τ ⊗ σ to ∗ D•[ρ ≤ τ](dτ ) · hτ, σ i. Using the formula for the boundary it can be shown this defines a morphism of complexes of sheaves. In case D• is projective Sh then εD• [ρ]: D•[ρ] ⊗ P• [ρ] → D•[ρ] is the identity in the first factor and contracts the second. It is therefore a homology isomorphism and by Lemma

2.1.28 εD• is a chain homotopy equivalence. The proof of the third part becomes a straightforward computation. Let ∗ P• be projective and consider any ρ ∈ X and (dσ ⊗ σ ) ∈ TP•[ρ], then

T(ε )[ρ] ∗ ∗ P• ∗ P ∗
∗ (dσ ⊗ σ ) / dσ ⊗ σ≤τ τ ⊗ τ ⊗ σ , id εTP• & u ∗ P ∗ ∗
dσ ⊗ σ≤τ τ ⊗ hτ, σ i

45 Remark 2.2.15. For any projective complex P• of compactly supported sheaves or cosheaves with values on Abf , the lemma above can be used to endow the chain complex Hom(TP•, P•) with an action of Σ2 defined to send f to (εP• ◦ Tf). This is an involution since

2 εP• ◦ T(εP• ◦ Tf) = εP• ◦ T f ◦ T(εP• ) = f ◦ εTP• ◦ T(εP• ) = f. Notation 2.2.16. Recall from Definition 1.1.5 that the arity 2 part of the operad S carries a free action of Σ2 and has the homology of a point. For any chain complex C with an action of Σ2, the chain complex of Σ2-equivariant maps from S(2) to C will be denoted CΣ2 . Let ϕ ∈ CΣ2 and as in Example 1.2.5, let (..., 2, 1, 2) be one of the degree d generators of S(2)•. The image of this generator via ϕ will be denoted ϕd, and if ϕ is a cycle then one has

d ∂ϕd+1 = (1 − (−1) T )ϕd.

Cycles in CΣ2 can be therefore thought of as homotopy fix points of the Σ2-action on C, compare with Example A.10. Remark 2.2.17. Applying Lemma 2.2.14 to the pair subdivision sheaf one Sh 2 • • gets that ε Sh : TP ∼ T C → C is a chain homotopy equivalence. P• • = • L ∗ Recall that by definition C [σ] is isomorphic to σ≤τ τ , the complex of cochains on the open star of σ. The concept of dual cone can be used to give another geometric interpretation for C•. Consider the amalgamation, described in Remark 2.2.13, of the close dual cone of a simplex σ into a subcomplex of the pair subdivision. The CW subcomplex corresponding to the open part of the dual cone of σ is parametrized by simplices τ satisfying τ ≥ σ, with a simplex of dimension k in the open star corresponds to a cell of dimension k − |σ| in the amalgamated open dual cone. For example,

stσ dc(σ) \ ∂dc(σ) −→

This geometric correspondence induces a chain homotopy equivalence be- tween C•[σ] and S−|σ|C•(dc σ, ∂ dc σ); the complex, suspended by |σ|, of rel- ative cochains on the closed dual cone modulo its boundary. Construction 2.2.18. The goal of this construction is to naturally obtain Sh Sh Σ2 a degree n cycle ϕ in HomSh(TP• , P• ) when the base X is a regular

46 n-dimensional pseudomanifold. This will be done taking the following steps. First, a morphism

|−| Sh Sh Σ2 D•[−] → HomSh(Σ TP• [−], P• [−]) of complexes of sheaves will be constructed. Second, a collection of com- 0 patible homomorphism C•(X ) → D•[σ] will be defined. third, the desired 0 cycle will be obtained by evaluating the fundamental cycle of C•(X ) in their composition. Let X be an ordered simplicial complex and X0 its barycentric subdi- vision. Let D•, as in Remark 2.2.13, be the projective complex of sheaves assigning to each σ in X the complex of simplicial chains in dc(σ), the dual cone of σ; and to each pair σ ≤ τ the inclusion C•(dc(τ)) → C•(dc(σ)). Each of this complexes is in a functorial manner an S-coalgebra and in particu- lar an S(2)-coalgebra, so by the hom-tensor adjunction the S(2)-structures define a morphism of complexes of sheaves
D•[−] → HomΣ2 S(2), D•[−] ⊗ D•[−] .

For every σ ∈ X one has

∼  • 0
 D•[σ] ⊗ D•[σ] = Hom C dc(σ) ,X \ dc(σ) , D•[σ] which is chain homotopy equivalent to

 •
 Hom C dc(σ) , ∂ dc(σ) , D•[σ] .

This complex is by Remark 2.2.17 chain homotopy equivalent to

|σ| Sh Sh
Hom Σ TP• [σ] , P• [σ] , so one gets a morphism of complexes of sheaves

|−| Sh Sh

D•[−] → HomΣ2 S(2), HomSh Σ TP• [−], P• [−] . (2.1)

For any chain in the simplicial chain complex of X0 one can construct an element in D• whose value in D•[σ] is computed by projecting the chain to the dual cone of the smallest vertex of σ, followed by taking its boundary and projecting it to the dual cone of the smallest edge of σ, and so on until

47 reaching the projection to the dual cone of σ; in symbols if σ = [v0, . . . , vn] one has

c 7−→ π[v0,...,vn] ◦ ∂ ◦ · · · ◦ π[v0,v1] ◦ ∂ ◦ π[v0](c) 0 with πσ denoting the projection from the simplicial chain complex of X onto D•[σ]. Notice that this construction decreases degree by |σ|. Let X be a simply connected regular pseudomanifold, i.e. an n-dimensional simplicial complex such that: each codimension 1 face is the boundary of exactly two distinct n-dimensional simplices, the boundary of the star of each simplex of codimension at least 2 is connected, and the sum of all n- dimensional simplices, denoted [X], is a cycle. Passing to the barycentric subdivision, let [X0] denote the cycle corresponding to [X]. The image of 0 [X ] in D• via the above construction is mapped by the morphism (2.1) to Sh Sh Σ2 Sh an n-dimensional cycle ϕ ∈ HomSh(TP• , P• ) . The pair (P• , ϕ) will be referred to as the visible symmetric complex of X.

2.3 Topological manifolds and S-comodules

In this section, as Theorem 2.3.4, the second of the two main technical re- sults of this work is presented. It states that the category of complexes of sheaves over an ordered simplicial complex X with values in Ab embeds as a differential graded full subcategory of the category of comodules over C•(X) as an S-coalgebra. This theorem is used to relate the algebraic surgery theory of Ranicki with comodules over E∞-coalgebras. In particular, Theorem 2.3.13 and Theorem 2.3.15 provide existence and uniqueness statements for homology manifold structures and topological manifold structures on the homotopy type of a Poincar´eduality regular pseudomanifold, using comodules on its S-coalgebra of chains.

Notation 2.3.1. The tensor product over X with the complex of cosheaf C• defines a functor from Sh(X, Ab)• to Ab•, see Definition 2.2.4 and Defini- tion 2.2.3 for unfamiliar terminology. For D• a complex of sheaves, D• ⊗X C• is given by M D•[σ] ⊗ C•[σ]/ ∼ σ∈X with dτ ⊗ C•[σ ≤ τ](cσ) ∼ D•[σ ≤ τ](dτ ) ⊗ cσ and differential graded structure induce from the tensor product of chain complexes. It is isomorphic

48 as abelian group to M D•[σ] ⊗ σ, σ∈X and elements of the form d ⊗ σ for some σ ∈ X will be referred to as canonical representatives of elements in D• ⊗X C•. Compare with the proof of Lemma 2.2.14. 0 Notice that given F : D• → D• a morphism of complexes of sheaves, the induced morphism is given in terms of canonical representatives by

f(d ⊗ σ) = F [σ](d) ⊗ σ.

Lemma 2.3.2. The functor − ⊗X C• lifts along the forgetful functor to the category of comodules over the S-coalgebra C•(X). Diagrammatically,

S coModC•(X) 7 forget  Sh(X, Ab)• / Ab• . −⊗C• X

Proof. For every σ ∈ X the complex C•[σ] = C•(cl σ, ∂) is an S-coalgebra naturally, so the functor C• : X → Ab• can be lifted along the forgetful S functor to C• : X → coModC•(X) with structure maps

⊗k ⊗(k−1) S(k) ⊗ C•[σ] → C•[σ] → C•[σ] ⊗ C•(X) .

The functor tensor product D• ⊗X C• inherits a S-comodule structure over C•(X) from its second factor.

Remark 2.3.3. By forgetting structure, D• ⊗X C• is also a comodule over C•(X) thought of as an S(2) coalgebra, i.e. a coalgebra over the operad generated by the arity 2 part of the operad S. The coaction of the generator (..., 1, 2, 1) of S(2) of degree k will be denoted by ∇k and, according to Lemma 2.3.2, it is defined for any class d ⊗ c ∈ D• ⊗X C• by

∇n(d ⊗ c) = d ⊗ ∆n(c), with the notation ∆n introduced in 1.2.9.

49 Theorem 2.3.4. The differential graded functor

S(2) − ⊗ C• : Sh(X, Ab)• −→ coModC (X) X • is full and faithful.

0 Proof. Let F : D• → D• be a morphisms of complexes of sheaves and as- sume the morphism induced by the functor − ⊗X C• is 0. Using canonical representatives, this implies that the abelian group homomorphism

M M M 0 (F [σ] ⊗ idσ): D•[σ] ⊗ σ −→ D•[σ] ⊗ σ σ∈X σ∈X σ∈X is 0, hence F [σ] = 0 for each σ ∈ X, i.e. F = 0. 0 Given an S-C•(X)-comodule map f : D• ⊗X C• → D• ⊗X C• one needs to construct a sheaf map inducing it. Let dσ ⊗ σ be a canonical representa- tive and write its image in terms of canonical representatives f(dσ ⊗ σ) = P 0 τ∈X dτ ⊗ τ. Since ∇n ◦ f = (f ⊗ id) ◦ ∇n one has for all n ≥ 0 that

f P 0 dσ ⊗ σ / τ∈X dτ ⊗ τ _ _ ∇n ∇n   d ⊗ ∆ σ / (?) ∼ P d0 ⊗ ∆ τ. σ n f⊗ id n τ∈X τ n

0 The above equation will be used to show that dτ = 0 for all τ 6= σ. Let n be 0 the largest |τ| so dτ 6= 0 and assume n > |σ|, then

X 0 (?)n = 0 ∼ dτ ⊗ τ ⊗ τ |τ|=n

0 so dτ = 0 for all τ of dimension n. Iterating this argument one has that 0 dτ = 0 for all τ of dimension greater than |σ|. For n = |σ| one has

X 0 X 0 (?)n = dτ ⊗ τ ⊗ σ ∼ dτ ⊗ τ ⊗ τ τ∈X |τ|=|σ|

0 so dτ = 0 for all τ 6= 0. For each σ ∈ X define the chain map

fσ : D•[σ] → D•[σ] 0 eσ 7→ eσ.

50 0 If the above collection of chain maps defines a morphism from D• to D• then it induces f, so it needs to be shown that for every ι : ρ → σ one has 0 D•[ι] ◦ fσ = fρ ◦ D•[ι]. Assume for an induction argument that this holds for all morphisms ι : ρ → σ with |σ| < n. For any simplex σ = [v0, . . . , vn] i denote its i-th face by σi = [v0,..., vˆi, . . . , vn] and ι : σi → σ. The induction assumption and the functoriality of sheaves imply that it suffices to show that 0 i i D•[ι ] ◦ fσ = fσi ◦ D•[ι ] (2.2) for all simplices σ of dimension n and i = 0, . . . , n. For any such σ ∈ X and d ∈ D•[σ] consider the following diagram

f d ⊗ σ / fσ(d) ⊗ σ (2.3) _ _ ∇0 ∇0   d ⊗ ∆ σ / (?) ∼ f (d) ⊗ ∆ τ. 0 f⊗ id σ 0 P Recall from Example 1.2.3 that ∆0[0, . . . , n] = i[0, . . . , i] ⊗ [i, . . . , n] so 0 0 projecting D• ⊗X C• ⊗ C•(X) onto D• ⊗X C• ⊗ [vn−1, vn] makes the equation in (2.3) be n
n f d ⊗ ι• σn ∼ fσ(d) ⊗ ι• σn which implies n
0 n
fσn ◦ D•[ι ] (d) = D•[ι ] ◦ fσ (d).

A completely analogous argument using T ∇0 verifies equation (2.2) for i = 0. For all 0 < i < n one consider the following diagram associated to ∇1

f d ⊗ σ / fσ(d) ⊗ σ (2.4) _ _ ∇1 ∇1   d ⊗ ∆ σ / (?) ∼ f (d) ⊗ ∆ τ. 1 f⊗ id σ 1 P Recall that ∆1[0, . . . , n] = i

51 Remark 2.3.5. One can consider the functor tensor product over X of the • cochain functor C and any E • ∈ coSh(X, A)•. The analogue of the results above can be proven by similar arguments, but they will not be used in this work. In particular, the category of complexes of cosheaves over X can be thought of as a full subcategory of the category of modules on the S-algebra of cochains on X.

L-theory for sheaf-like S-comodules Definition 2.3.6. (Sheaf-like comodules and duality) An S-comodules D over C•(X) is said to be sheaf-like if it is isomorphic to one of the form D• ⊗XC• with D• a projective complex of sheaves with values on the cat- egory Abf of finitely generated abelian groups. The Ranicki duality func- tor, Definition 2.2.9, induces by Lemma 2.3.4 a contravariant functor on ∼ the subcategory of sheaf-like comodules. Explicitly, let D = D• ⊗XC• and 0 ∼ 0 0 D = D• ⊗XC• be sheaf-like and f ∈ HomcoMod(D,D ). By Lemma 2.3.4 0 there exists F ∈ HomSh(D, D ) so that F ⊗XC• = f. Define, abusing nota- tion, Tf :TD0 → TD to be

0 (TF ) ⊗XC• : (T D ) ⊗XC• → (T D) ⊗XC• . 2 Remark 2.3.7. The natural transformation ε : T → idSh of Lemma 2.2.14 induces an analogous natural transformation for the duality of sheaf-like comodules. In particular, the chain complex HomcoMod(TD,D) has an action of Σ2 given by f 7→ εD ◦ T f. For any sheaf-like comodule D = D ⊗XC•, Theorem 2.3.4 gives a chain isomorphism

Σ2 ∼ Σ2 HomSh(T D, D) = HomcoMod(TD,D) . See 2.2.16 for the definition of these complexes. Definition 2.3.8. (Connective sheaf-like and Poincar´ecomodules) A sheaf- like comodule is said to be connective if it is chain homotopy equivalent, as S-comodule, to one which equals 0 in negative degrees. An n-dimensional weak Poincar´ecomodule is a pair (D, ϕ) with D a Σ2 connective sheaf-like comodule and ϕ a cycle of degree n in HomcoMod(TD,D) such that ϕ0 is a homology isomorphism. (See Remark 2.2.16 for unfamiliar notation.) An n-dimensional strong Poincar´ecomodule is an n-dimensional weak Poincar´ecomodule (D, ϕ) so that ϕ0 is a chain homotopy equivalence of S-comodules.

52 Notation 2.3.9. Let f : C → C0 be a chain map between chain complexes. Denote the mapping cone of f by Cone(f), i.e. the chain complex ΣC ⊕C0 with boundary defined by (Σ(c) + c0) 7→ Σ(∂(c)) + f(c) + ∂(c0), where Σ(−) stands for suspension.

Definition 2.3.10. (Cobordism) A weak cobordism between n-dimensional weak Poincar´ecomodules (D, ϕ) and (D0, ϕ0) consists of a connective sheaf- Σ2 like comodule E with a degree (n + 1)-chain φ ∈ HomcoMod(TE,E) and a couple of maps f : D → E and f : D0 → E satisfying:

1. ∂φ = (f ◦ ϕ ◦ Tf) − (f 0 ◦ ϕ0 ◦ Tf 0).

def 0 0
0 2.( φ/ϕ)0 = φ0 + (ϕ0 ◦ Tf) + (ϕ0 ◦ Tf ) :TE → Cone(f ⊕ (−f )) is a homology isomorphism.

A strong cobordism between n-dimensional strong Poincar´ecomodules 0 0 (D, ϕ) and (D , ϕ ) is a weak cobordism between them such that (φ/ϕ)0 is a chain homotopy equivalence of S-comodules.

Example 2.3.11. The central example of a weak Poincar´ecomodule for the applications of this work comes from Construction 2.2.18. Let X be a simply-connected regular pseudomanifold which is an n-dimensional Poincar´e duality space. By Theorem 2.3.4 and Remark 2.3.7 the image by − ⊗XC• of the visible symmetric complex, denoted (P, ϕP), would be an n-dimensional weak Poincar´ecomodule if (ϕP)0 induces a homology isomorphism. To see this is the case, recall that by definition one has for any complex L of sheaves D• that D• ⊗XC• = σ∈X D•[σ] ⊗ C•[σ]/∼ and C•[σ] = C•(cl σ). Contracting each second factor one gets a chain map inducing an isomorphism in homology M D• ⊗XC• → D•[v]/∼ˆ (2.5) v∈X(0) 0 with D•[v ≤ σ](dv)∼ ˆ D•[v ≤ σ](dv0 ) for every σ ∈ X. This assignment is functorial with a morphism of complexes of sheaf F : D → D0 inducing the L chain map v∈X(0) F [v]. Sh Recall from Remark 2.2.13 that P• [σ] is chain homotopy equivalent to the simplicial chains on the closed dual cone of σ, in symbols

Sh che P• [σ] → D•[σ] = C•(dc(σ));

53 Sh and from Remark 2.2.17 that TP• [σ] is chain homotopy equivalent to the relative simplicial cochains suspended by |σ| of the closed dual cone of σ modulo its boundary, in symbols

Sh che |σ| • TP• [σ] → Σ C (dc(σ), ∂ dc(σ)). By the previous two observations, the morphism (2.5) and the definition of ϕP one has a commutative diagram

Sh (ϕP)0 Sh TP• ⊗ C• / P• ⊗ C• X X

•  0  0 C (X ) / C•(X ) −∩[X0] with vertical arrows representing homology isomorphisms. Since X0 is a Poincar´eduality space, the morphism − ∩ [X0] induces an isomorphism in homology with a degree shift of n and therefore, the morphism (ϕP)0 does as well. Definition 2.3.12. An ANR homology n-manifold is a finite dimensional absolute neighborhood retract X satisfying for every x ∈ X ( Z if i = n Hi(X,X \{x}) = 0 if i 6= n.

Theorem 2.3.13. Let X be a simply-connected regular pseudomanifold which is an n-dimensional Poincar´eduality space with n > 4. For any homotopy equivalence between an ANR homology n-manifold and X, there exists a weak cobordism between a strong n-dimensional Poincar´e comodule and (P, ϕP). Conversely, for every weak cobordism between a strong n-dimensional Poincar´ecomodule and (P, ϕP), there exists a homotopy equiv- alence between X and an ANR homology n-manifold. Two such homotopy equivalences are related by an h-cobordism relative to boundary if and only if their corresponding strong Poincar´ecomodules are related by a strong cobordism. In order to obtain existence and uniqueness statements for topological manifold structures on X one needs to “tame the fundamental group” of the weak cobordisms involved.

54 Definition 2.3.14. (Admissible cobordisms) Using the notation of Defini- tion 2.3.10, a weak cobordism is said to be admissible if the the mapping cone of (φ/ϕ)0 is chain homotopy equivalent as S-comodule to a sheaf-like comodule which equals 0 in degrees less than or equal to 1.

Theorem 2.3.15. Let X be a simply-connected regular pseudomanifold which is an n-dimensional Poincar´eduality space with n > 4. For any homotopy equivalence between a topological n-manifold and X, there exists an admissible weak cobordism between a strong n-dimensional Poincar´ecomodule and (P, ϕP). Conversely, for every admissible weak cobor- dism between a strong n-dimensional Poincar´ecomodule and (P, ϕP), there exists a homotopy equivalence between X and a topological n-manifold. Two such homotopy equivalences are related by an h-cobordism relative to boundary if and only if their corresponding strong Poincar´ecomodules are related by a strong cobordism.

Remark 2.3.16. Starting with a homotopy equivalence in either of the the- orems above, the weak cobordism obtained has the further property that the morphism from each of its boundary components induces an isomorphism in homology. Also, in both theorems above, given a strong Poincar´ecomplex weak cobordant to (P, ϕP), there exist another strong Poincar´ecomplex, which is strong cobordant to the original one, and a weak cobordism between it and (P, ϕP) such that the morphism from each of its boundary components induces and isomorphism in homology.

Proof of Theorem 2.3.13 and Theorem 2.3.15. Both of the proofs are obtain by relating to the algebraic surgery theory as developed by Ranicki in [30]. The differential graded category of connective sheaf-like comodules over C•(X) with duality T is, by Theorem 2.3.4, equivalent to the category of connective complexes of projective sheaves over X with Ranicki duality. This category is equivalent to the category of connective complexes of X-based modules defined in [30, p.63], see [33, p.169] for a proof, with the duality defined in [30, p.75]. The weak Poincar´ecomodule (P, ϕP) represents, under the above identi- fication, the “(1/2)-visible symmetric signature” of Ranicki, see Remark 16.8 [30, p.181]. Also under this identification, weak cobordisms, admissible weak cobordism and strong cobordisms correspond to symmetric cobordisms in

55 the algebraic bordism categories Λh0i(Z,X), Λh1/2i(Z,X) and Λh0i(Z)∗(X) respectively, as defined in pages 157, 164 and 158 of [30]. The statements now follow from Theorem 17.4, Theorem 18.5, Proposi- tion 25.7 and Remark 25.13 of [30].

56 Appendix A

Categorical Background

In this appendix the notions of limits, colimits and Kan extensions are col- lected, emphasizing their use in constructions associated to simplicial sets.

Definition A.1. (Diagrams) A diagram in C indexed by I is a functor I → C with I a small category.

Example A.2. (Constant diagrams) For any small category I and object c in a category C, define the constant diagram indexed by I with value c by declaring the image of any objet in I to be c and of any morphism to be idc. Example A.3. Let G be a group and G be the category with one object ∗ ∼ and HomG(∗, ∗) = G. A diagram G → C is the same data as and object in C with a G action.

Limits

Definition A.4. (Limits) Let D : I → C be a diagram. The limit of D consists of an object limI D in C and a natural transformation ϕ from the constant diagram indexed by I with value limI D to D, satisfying the fol- lowing universal property. For any constant diagram indexed by I provided with a natural transformation φ to D, there exists a unique morphism f from its constant value cone to limI D such that for every i ∈ I one has

57 ϕ(i) ◦ f = φ(i). Diagrammatically,

D(i→j) D(i) / D(j) T a ϕ(i) ϕ(j) = J

limI D φ(i) O φ(j) f cone commutes for every (i → j) ∈ HomI (i, j). Example A.5. (Orbits) Let C be a small category. Recall from Example A.3 that an object in C with an action of a group G can be thought of as a diagram D : G → C. Denote D(∗) = X ∈ C and notice that for any g ∈ G one has the commutative diagram

g X / X . _ ?

limGD The universal property of limits implies that the limit of D equals the set of orbits of the action, i.e. colimG D = XG. Example A.6. (Initial objects, products and equalizers) Consider a diagram in C with index category one of the following:

Λ z }| { ( a) ∅ b) •• ... • c) • 6 • . If the limit of the diagram exists it is called respectively a) The initial object. For example, the empty topological space or the zero abelian group. Q b) The product, which is denoted Λ or × ... ×. For example, cartesian product of topological spaces or tensor product of abelian groups. c) The equalizer, which is denoted eq(−). For example, for spaces one has f coeq(X ⇒ Y ) = {x ∈ X : f(x) = g(x)}. For abelian groups, the equalizer g of a pair of maps where one of them is the zero map equals the kernel of the other map.

58 The following statement, whose proof can be found in [18, p.112], shows that a category where all products and equalizers exist is such that the limit of any diagram exists. Such categories are called complete.

Lemma A.7. Let D : I → C be an diagram in a complete category C. The limit of D is given by  Y Y  eq D(i) ⇒ D(j) i i→j where one of the maps comes from projecting to the source of each morphism and then applying the corresponding morphism induced by D, while the other is induced from directly projecting to the target of each morphism.

Example A.8. (Pullbacks) The limit of a diagram in a cocomplete small category of the form • C −→ g  f  • / • A / B is according to Lemma A.7 equal to

 p3×p3  eq A × C × B ⇒ B × B = {(x, y, b): f(x) = g(y) = b}. f◦p1×g◦p2 ∼ This limit will be denoted by A ×B C = {(x, y): f(x) = g(y)} and referred f g to as the pullback of A → B ← C.

Colimits

Definition A.9. (Colimits) Let D : I → C be a diagram. The colimit of D consists of an object colimI D in C and a natural transformation ϕ from D to the constant diagram indexed by I with value colimI D, satisfying the following universal property. For any constant diagram indexed by I pro- vided with a natural transformation φ to D, there exists a unique morphism f from colimI D to its constant value cocone such that for every i ∈ I one

59 has φ(i) = f ◦ ϕ(i). Diagrammatically,

D(i→j) D(i) / D(j) ϕ(i) ϕ(j) # { colim D φ(i) I φ(j) f cocone  ~ commutes for every (i → j) ∈ HomI (i, j). Example A.10. (Fix points) Let C be a small category. Recall from Exam- ple A.3 that an object in C with an action of a group G can be thought of as a diagram D : G → C. Denote D(∗) = X ∈ C and notice that for any g ∈ G one has the commutative diagram

g X / X .

! } colimG D The universal property of colimits implies that the colimit of D equals the G fix point set of the action, i.e. colimG D = X . Example A.11. (Terminal objects, coproducts and coequalizers) Consider a diagram in C with index category one of the following:

Λ z }| { ( a) ∅ b) •• ... • c) • 6 • . If the colimit of the diagram exists it is called respectively a) The terminal object. For example, the topological space with one ele- ment or the zero abelian group. ` b) The coproduct, which is denoted Λ or t ... t. For example, disjoint union of topological spaces or direct sum of abelian groups. c) The coequalizer, which is denoted coeq(−). For example, for spaces one f . has coeq(X ⇒ Y ) = Y f(x) ∼ g(x). For abelian groups, the coequalizer g of a pair of maps where one of them is the zero map equals the cokernel of the other map.

60 The following lemma shows that a category where all coproducts and co- equalizers exist is such that the colimit of any diagram exists. Such categories are called cocomplete.

Lemma A.12. Let D : I → C be an diagram in a cocomplete category C. The colimit of D is given by  a a  coeq D(i) ⇒ D(i) i→j i where one of the maps comes from the identity D(i) → D(i), while the other comes from the morphism D(i) → D(j) induced by D from the morphisms i → j. Proof. This is a variation of the proof of the analogue statement for limits in Lemma A.7.

Example A.13. (Pushouts) The colimit of a diagram in a cocomplete small category of the form f • / • B / C

−→ g   • A is according to Lemma A.12 equal to

 id t id   coeq B t B ⇒ A t B t C = A t C f(b) ∼ g(b). ftg

This colimit will be denoted by A tB C and referred to as the pushout of g f A ← B → C.

Kan extensions

Definition A.14. (Kan extensions) Let F : C → A and E : C → B be a pair of functors. The right Kan extension of F along E is a functor RanE F : B → A and a natural transformation φ : F → RanE F ◦E satisfying the following universal property. For any pair R : B → A and ψ : F → R ◦E

61 there exists a unique θ : R → RanE F such that ψ = φ ◦ θF with θF (c) = θ(F (c)) for all c ∈ C. Diagrammatically,

F C / A ? S CK φ E RanE F [c  θ B R

The left Kan extension of F along E is a functor LanE F : B → A and a natural transformation φ : LanE F ◦ E → F satisfying the following universal property. For any pair R : B → A and ψ : R ◦E → F there exists a unique θ : LanE F → R such that ψ = θF ◦ φ with θF (c) = θ(F (c)) for all c ∈ C. Diagrammatically,

F C / A ? S φ E Ó LanE F

 θ # B R Kan extensions need not exist. But if A is cocomplete then one can prove the existence of the right Kan extension by exhibiting a formula. Cor- respondingly, if A is complete then the left Kan extension exists and it is also given by a formula, both of which are presented in Lemma A.16. One begins by defining the following category. Definition A.15. (Comma category) Let A →S C ←T B be a diagram of categories. The comma category (S ↓ T ) has objects all triples (h, a, b) with S(a) 7→h T (b) and morphisms (f, g):(h, a, b) → (h0, a0, b0) all pairs satisfying h S(a) / T (b)

S(f) T (g)  h0  S(a0) / T (b0). When A is the category with one object ∗ and one morphism and S(∗) = c, the category (S ↓ T ) will be denoted simply by (c ↓ T ). The category (S ↓ c) is similarly defined.

62 The following statement and a proof can be found in [18, p.237] and [18, p.244]. Lemma A.16. Let F : C → A and E : C → B be a pair of functors. 1. If A is cocomplete then for any b ∈ B

RanE F (b) = colim F. (E↓ b)

2. If A is complete then for any b ∈ B

LanE F (b) = lim F. (b ↓ E)

Definition A.17. (Simplicial category and simplicial sets) Let ∆ denote the simplicial category whose objects are the finite non-empty totally ordered sets, commonly denoted [0, 1, . . . , n], and whose morphisms are the order preserving functions. Any such function can be obtained as a composition of basic ones, called cofaces and codegeneracies, which insert or delete a single element. The cofaces and codegeneracies will be respectively denoted di :[n − 1] → [n] and si :[n + 1] → [n] and they satisfy well know relations. Define the category of simplicial sets to be op s Set = HomCat(∆ , Set), the category of contravariant functors from the simplicial category to the category of sets. For X ∈ sSet, the image of [0, . . . , n] will be denoted Xn and referred to as the set of n-simplices of X. Simplices which are the image of lower dimensional simplices are said to be degenerate and a simplicial set is said to be n-dimensional if there exists a non-degenerate n-simplex and for m > n all m-simplices are degenerate. Remark A.18. (Yoneda) There exists a full and faithful functor from the simplicial category ∆ into the category of simplicial sets given on objects by ∆ −→ s Set [0, . . . , n] 7−→ Hom∆([0, . . . , n], −). Such functor will be called the Yoneda embedding and the images of n [0, . . . , n] ∈ ∆ will be denoted by ∆ . Notice that for any X• ∈ s Set one has n Homs Set(∆ ,X•) = Xn, a fact referred to as Yoneda lemma.

63 Definition A.19. (Realization and nerve) Let C be a cocomplete category and F : ∆ → C a functor. Denote the Yoneda embedding ∆ → s Set by Y. The realization with respect to F is the right Kan extension of F along Y. The nerve with respect to F is the functor defined on objects
by c 7−→ [0, . . . , n] 7→ HomC(F [0, . . . , n], c) . Diagrammatically,

F ∆ /5 C Y  x s Set . Remark A.20. It is a theorem of Daniel Kan [11] that the realization and nerve with respect to a functor form a universal adjoint pair. Example A.21. (Geometric realization and singular complex) Consider the embedding ∆ → Top of the simplicial category into the category of topologi- cal spaces sending [0, . . . , n] to the standard topological n-simplex |∆n|. The realization with respect to this functor of any X• ∈ s Set can be described by Lemma A.16 and Lemma A.12 as the coequalizer of

a n a n |∆ | ⇒ |∆ | n ∆ n  ∆ → X• X• " ∆k H or equivalently using the Yoneda lemma as

 a n a n  coeq Xn × |∆ | ⇒ Xn × |∆ | , n≥0 n≥0 which utilizing the cofaces and codegeneracies di : [0, . . . , n − 1] → [0, . . . , n] and si : [0, . . . , n + 1] → [0, . . . , n] in ∆ can be expressed as ∗ a n . di x × p ∼ x × di ∗p Xn × |∆ | ∗ s x × p ∼ x × si ∗p. n≥0 i

This topological space will be called the geometric realization of X• and will be denoted by |X•|. In this context, the nerve of a topological space X will be called the singular simplicial complex of X and it is the simplicial set Sing•(X) given by n [0, . . . , n] 7→ HomTop(|∆ |,X).

64 Example A.22. (Normalized chain complex) Consider the embedding ∆ → Ab• of the simplicial category into the category of chains complexes sending n [0, . . . , n] to the standard chain complex C•(∆ ). As in the previous example the realization with respect to this functor of any X• ∈ s Set can be described as M . d ∗x ⊗ c ∼ x ⊗ d c X ⊗ C (∆n) i i ∗ n • s ∗x ⊗ c ∼ 0. n≥0 i

This chain complex will be called the normalized chain complex of X•. Example A.23. (Nerve of a category) Consider the embedding ∆ → Cat of the simplicial category into the category of small categories sending [0, . . . , n] to the category n with one object for each i ∈ {0, 1, ..., n} and one morphisms i → j whenever i ≤ j. The nerve of a category C ∈ Cat is the simplicial set N•(C) defined as the nerve with respect to this functor. Explicitly N•(C) is given by [0, . . . , n] 7→ HomCat(n, C).

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